The local-to-global property for Morse quasi-geodesics

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.


Introduction
We provide a new approach for studying groups and spaces with features of non-positive curvature by defining a local-to-global property for Morse quasi-geodesics. We show a variety of results from hyperbolic spaces effortlessly extend to any space satisfying this Morse local-to-global property. We prove examples of such spaces include the mapping class group and Teichmüller space of finite type, orientable surfaces, the fundamental groups of closed 3-manifolds, and all CATp0q spaces.
1.1. The local-to-global property for Morse quasi-geodesics. Morse quasi-geodesics generalize a key property of quasi-geodesics in hyperbolic spaces. Given a function M : r1, 8qˆr0, 8q Ñ r0, 8q, a quasi-geodesic γ is M -Morse if every pλ, ǫq-quasi-geodesic with endpoints on γ is contained in the M pλ, ǫq-neighborhood of γ. We call the function M the Morse gauge of γ. 1 The study of Morse quasi-geodesics arose from trying to understand the "hyperbolic directions" in a non-hyperbolic space [CS15] and has since received immense interest in the literature (see [ABD21,HH18,ACGH17,DMS10,OsOS09] for a sampling). Numerous results from hyperbolic spaces have fruitful generalizations to spaces containing infinite Morse quasi-geodesics, particularly with respect to the study of stable subgroups [AMST19, CH17, ADT17] and the quasi-isometric classification of spaces [Cor17,CCM19]. Our new contribution to the study of Morse quasi-geodesics is the introduction of a local-to-global property for Morse quasi-geodesics.
Date: August 4, 2021. 1 To simplify proofs, we use a more technical, but equivalent, definition of Morse quasi-geodesics; see Section 2.2.
satisfying a minor technical condition that encompasses these examples. The proof for hierarchically hyperbolic spaces rests upon a characterization of Morse quasi-geodesics in these space due to Abbott, Behrstock, and Durham [ABD21]. Virtually solvable groups and groups with infinite order central elements are examples of spaces where the Morse geodesics have uniformly bounded length. We call these spaces Morse limited and prove they all trivially satisfy Definition B by taking L sufficiently larger than the bound on the lengths of the Morse geodesics. These "trivial" examples are essential to proving that the fundamental group of any closed 3-manifold has the Morse local-to-global property. The fundamental group of a 3-manifold decomposes into a free product of groups that are either virtually solvable or hierarchically hyperbolic spaces. The proof for all closed 3-manifold thus follow from the above plus the next theorem about relatively hyperbolic groups.
Theorem E (Hyperbolic relative to Morse local-to-global). Let G be a group hyperbolic relative to the subgroups H 1 , . . . , H n . If each H i has the Morse local-to-global property, then G has the Morse local-to-global property.
The proof of Theorem E is, somewhat surprisingly, the most involved of this paper. Our proof develops a theory of "deep points" for local quasi-geodesics in a relatively hyperbolic space analogous to the deep points for geodesics introduced by Hruska [Hru10]. The deep points partition a local quasi-geodesic into pieces that alternatingly avoid and pass through the peripheral subsets of the relatively hyperbolic space and maybe of independent interest in the study of relative hyperbolicity.
Theorem F (Local quasi-geodesics in relatively hyperbolic spaces). Let X be hyperbolic relative to a collection P of peripheral subsets. For each λ ě 1 and ǫ ě 0, there exists L ą 0 so the following holds. For every pL; λ, ǫq-local quasi-geodesic γ in X, there exist peripheral subsets tP 1 , . . . , P n u and a decomposition γ " σ 0˚α1˚σ1˚¨¨¨˚αn˚σn so that each α i is contained in a uniform neighborhood of P i and each σ i uniformly quasi-isometrically embeds in the coned-off space p X. This implies each σ i is uniformly a Morse quasi-geodesic in X.
1.3. Consequences of the Morse local-to-global property. Once the difficult task of showing a space satisfies the Morse local-to-global property is completed, many arguments from hyperbolic spaces can be seamlessly generalized.
Our first examples of this are extensions of Gitik's combination theorems for quasiconvex subgroups of hyperbolic groups to stable subgroups of Morse local-to-global groups. A streamlined version of the combination theorems is the following (see Section 3.1 for complete statements).
Theorem G (Combinations of stable subgroups). Let G be a finitely generated group with the Morse local-to-global property and P , Q be stable subgroups of G. For each finite generating set of G, there exists C ą 0 (depending only on the stability parameters of P and Q) so that if P X Q contains all elements of P Y Q whose length in G is at most C, then the subgroup xP, Qy is stable in G and isomorphic to P˚P XQ Q. Further, if P is malnormal in G, then the same conclusion holds if we only require P X Q to contain all elements of P whose length in G is at most C.
Stable subgroups are a strong generalization of quasiconvex subgroups that requires every pair of elements to be joined by a uniform Morse geodesic that stays uniformly close to the subgroup. Stable subgroups were introduced by Durham and Taylor to study the convex cocompact subgroups of the mapping class group [DT15], but have sense been studied in a variety of non-hyperbolic groups [HH18,ABD21,Tra19]. Theorem G is particularly interesting in the mapping class group, where it produces combination theorems for the convex cocompact subgroups by the work of Durham and Taylor (see Section 1.4 for details). Similar combination theorems have been proved by Dey, Kapovich, and Leeb for Anosov subgroups of semi-simple Lie groups [DKL19] and by Martínez-Pedroza and Sisto for relatively quasiconvex subgroups of relatively hyperbolic groups [MPS12].
Theorem L is similar to Gromov's local condition for hyperbolicity in simply connected spaces [Gro87, Theorem 4.1.A]. While our result does not apply to all simply connected spaces, the proof of Theorem L is considerably shorter than proofs of Gromov's condition appearing in the literature (see [Cou14,Bow91,Pap96]) while still capturing the important cases of CATp0q spaces and the universal covers of closed 3-manifolds.
1.4. Convex cocompact subgroups. Farb and Mosher originally introduced convex cocompact subgroups of the mapping class group by analogy with convex cocompact subgroups of Kleinian groups [FM02]. These subgroups have since attracted substantial attention as they play a critical role in the theory of extensions of hyperbolic groups and the geometry of surface bundles [Ham,KL08]. A theorem of Durham and Taylor established that the stable subgroups of the mapping class group are precisely the convex cocompact subgroups [DT15]. Using this equivalence, Theorem G produces a combination theorem for convex cocompact subgroups of the mapping class group.
Theorem M. Let MCGpSq be the mapping class group of a orientable surface S of finite type. Suppose P and Q are convex cocompact subgroups of MCGpSq. There exists C ą 0 so that if P X Q contains all elements of P Y Q whose word length in MCGpSq is at most C, then the subgroup xP, Qy is convex cocompact and isomorphic to P˚P XQ Q. Further, if P is malnormal, then the same conclusion holds if we only require P X Q to contain all elements of P whose length in MCGpSq is at most C. Since examples of virtually free convex cocompact subgroups are abundant, a strategy to answer Question N could involve using Theorem M to create new examples using an amalgamated free product. This remains beyond our current reach in the mapping class group, but we produce new examples of one-ended stable subgroups of CATp0q groups using our combination theorem in Example 3.4.
1.5. Outline of the paper. Section 2 consists of preliminary results on Morse quasi-geodesics and the definition of the Morse local-to-global property. Section 3 is dedicated to consequences of the Morse local-to-global property. Sections 4 and 5 contain proofs that the Morse local-to-global property is exhibited by a large number of spaces. Section 4 contains the cases of CATp0q spaces, hierarchically hyperbolic spaces, and virtually solvable groups, while Section 5 contains the cases of relatively hyperbolic spaces and 3-manifold groups.
Below, we report open questions that arose from our study of the Morse local-to-global property. This outlines a rich new direction for research in groups with features of non-positive curvature.
1.6. Open questions and further directions. The examples we give of groups without the Morse local-to-global property are all infinitely presented. Our first question, inspired by Ruth Charney, asks how "nice" such a group could be. Question 1. Does there exist an example of a finitely presented group that is not Morse local-toglobal? Does there exist an example of a group with quadratic Dehn function that is not Morse local-to-global? Does there exist an example of a bi-automatic group that is not Morse local-toglobal?
We have shown that several different notions of non-positive curvature (CATp0q, hierarchically hyperbolic, relatively hyperbolic) imply the Morse local-to-global property. It is natural to wonder if other groups with features of non-positive curvature are also Morse local-to-global.  Several of the properties of Morse local-to-global groups that we establish are reminiscent of properties of acylindrically hyperbolic groups. Further, every known example of a Morse local-toglobal group that is not Morse limited or virtually cyclic is acylindrically hyperbolic. We ask if this is always the case.
Question 5. If G is a Morse local-to-global group that is not Morse limited or virtually cyclic, is G acylindrically hyperbolic?
Note, the converse of Question 5 is false. Counterexamples can be found by taking the free product of two copies of either of the groups we show to not have the Morse local-to-global property in Example 4.10. The resulting group is acylindrically hyperbolic, but not Morse local-to-global.
One consequence of a positive answer to Question 5 would be that all Morse local-to-global groups that are not Morse limited or virtually cyclic would have property P naive [AD19]. That is, for any collection of group elements g 1 , . . . , g n , there would exist an element h so that xg i , hy -xg i y˚xhy for each 1 ď i ď n. Property P naive is a strong version of the ping-pong lemma and very close to several of the results in this paper, making it an attractive property to study directly in these groups.
Question 6. If G is a Morse local-to-global group that is not Morse limited or virtually cyclic, does G have property P naive ?
A classical application of the local-to-global property of quasi-geodesic in hyperbolic groups is Cannon's proof that the geodesics of a hyperbolic group form a regular language [Can84]. In [EZ18], the local nature of Morse geodesics in CATp0q spaces is used to show that the M -Morse geodesics of a CATp0q group also form a regular language. Extending these result to all Morse local-to-global groups would produce new results for both the mapping class group and 3-manifold groups.
Question 7. Do the M -Morse geodesics of a Morse local-to-global group form a regular language? 2 In CATp0q spaces and hierarchically hyperbolic spaces, Morse quasi-geodesics have equivalent formulations in terms of contracting and divergence properties [CS15,ABD21,RST18]. The authors of [ACGH17] showed these characterizations do not hold in general metric spaces, but provided characterization of Morse quasi-geodesics using much weaker contracting and divergence properties. We ask if Morse quasi-geodesics in Morse local-to-global spaces have stronger contracting and divergence properties as in the CATp0q and hierarchically hyperbolic cases. A positive resolution of this question may assist in answering Questions 5 and 7.
Question 8. Do Morse quasi-geodesic in Morse local-to-global spaces have stronger contracting or divergence properties than general Morse quasi-geodesics?
In section 4.1, we show that if a space has any asymptotic cone with no cut-points, then it is Morse limited. It is unknown if the converse is true.
Question 9. If G is Morse limited, does G have an asymptotic cone with no cut-points?
A positive answer to Question 9 would upgrade Corollary I to say every Morse local-to-global group either contains a Morse element or has an asymptotic cone with no cut-points. This would give a positive answer to [BD14,Question 6.10] that asks if every CATp0q group with a cut-point in every asymptotic cone must contain a Morse quasi-geodesic.
Our final two questions seek to improve on results in is paper. The first asks to make our stable subgroup combination theorem effective in specific examples.
Problem 10. Effectivize the stable subgroup combination theorems (Theorem 3.1) for the mapping class group and/or CAT(0) groups.
In the case of the mapping class group, Bowditch shows a much stronger version of Theorem K, namely that the set of translation lengths of all Morse elements, independent of Morse gauge, is a discrete set of rational numbers [Bow08, Corollary 1.5]. We ask if the same improvement can be made for all Morse local-to-global groups.
Question 11. If G is Morse local-to-global group, is the set of translation lengths of all Morse elements of G a discrete set of rational numbers?
Acknowledgments: The authors are grateful to Sam Taylor for insightful conversations and answering many questions. They also thank Emily Stark for helping to find references for Example 3.4 and Thomas Ng for comments on a draft of this paper. The first two authors thank their advisors, Jason Behrstock and Alessandro Sisto, for their support and guidance during this project. The first and third author thank the organizers of the 2019 Tech Topology conference where some of the work on this project was completed. The first author would also like to thank the FIM Institute for Mathematical Research at ETH Zürich for their hospitality during the conference "Groups, spaces, and geometries" where much of the work on this project was completed. Lastly, the authors thank the anonymous referee for their helpful comments.

Background and preliminaries
2.1. Groups and Cayley graphs. The majority of the metric spaces we are interested in will be the Cayley graphs of finitely generated groups.
Definition 2.1. If G is a group generated by a set S, then the Cayley graph of G with respect to S is the graph with all elements of G as vertices and where g, h P G are joined by an edge if g´1h P S Y S´1. We denote the Cayley graph of G with respect to S by CaypG, Sq.
Given a fixed generating set S for G, we use |g| to denote the minimum number of elements of S Y S´1 needed to write g, i.e., the word length of g with respect to S. The word length of g with respect to S is equal to the distance in CaypG, Sq from the identity e to g. Every path is CaypG, Sq is labeled by some word w in the set S Y S´1. If a path labeled with the word w starts at e and ends at g, then the word w represents g. Throughout this article, we will always implicitly consider a finitely generated group as a metric space by equipping it with a word metric with respect to some finite generating set.

2.2.
Morse quasi-geodesics. In this section, pX, dq will denote a metric space and I will denote a closed, but possibly unbounded, interval of R. Throughout this article, if γ : I Ñ X is a map, we will abuse notation by using γ to refer the image of γ in X. The main objects of this paper are quasi-geodesics with a stability property called Morse. Definition 2.2 (Quasi-geodesics). A map γ : I Ñ X is a pλ, ǫq-quasi-geodesic if λ ě 1, ǫ ě 0 and for each s, t P I 1 λ |s´t|´ǫ ď d`γpsq, γptq˘ď λ|s´t|`ǫ.
We say γ is a finite quasi-geodesic if I is a compact interval and an infinite quasi-geodesic if I is not compact. A subsegment of a quasi-geodesic γ : I Ñ X is a restriction of γ to a closed, connected subset of I. The parametrized length of a quasi-geodesic is the length of the domain of the quasi-geodesic. Definition 2.3 (Morse quasi-geodesic). Let M : r1, 8qˆr0, 8q Ñ r0, 8q be a function. The quasigeodesic γ : I Ñ X is an M -Morse quasi-geodesic if for all s ă t in I, if α is a pλ, ǫq-quasi-geodesic with endpoints γpsq and γptq, then the Hausdorff distance between α and γ| rs,ts is bounded by M pλ, ǫq. If γ is an M -Morse pλ, ǫq-quasi-geodesic, we say γ is an pM ; λ, ǫq-Morse quasi-geodesic.
Those familiar with the literature will note that Definition 2.3 is stronger than the usual definition of a Morse quasi-geodesic. The next lemma shows that, up to a modification of the Morse gauge, Definition 2.3 is equivalent to the usual definition of Morse quasi-geodesic. This allows us to simplify proofs by utilizing the stronger definition of Morse when working with Morse quasi-geodesics, but only demonstrating the weaker condition in Lemma 2.4 when proving a quasi-geodesic is Morse.
Lemma 2.4 (Verification of Morse). Let γ : I Ñ X be a pλ, ǫq-quasi-geodesic. Suppose there exists a function N : r1, 8qˆr0, 8q Ñ r0, 8q such that for all s, t P I, if α is a pλ 1 , ǫ 1 q-quasi-geodesic with endpoints γpsq and γptq, then α is contained in the N pλ 1 , ǫ 1 q-neighborhood of γ. The trademark of Morse quasi-geodesics is that they mimic the behavior of quasi-geodesics in a hyperbolic space. In particular, triangles and rectangles with Morse sides will be slim and every Morse quasi-geodesic will be close to a Morse geodesic. Definition 2.6. A geodesic triangle (quadrilateral) is δ-slim, if each side is contained in the δneighborhood of the other two (three) sides. ‚ If T is a geodesic triangle in X where two of the three sides are M -Morse, then T is 4M p3, 0q-slim and there exists a Morse gauge M 1 " M 1 pM q such that all three sides of T are M 1 -Morse. ‚ If R is a geodesic quadrilateral in X where three of the four sides are M -Morse, then R is 8M p3, 0q-slim and there exists a Morse gauge M 1 " M 1 pM q such that all four sides of R are M 1 -Morse.
Proof. This follows immediately from Definition 2.3 The last result we record on Morse geodesics is an application of the Arzela-Ascoli Theorem to say limits of Morse geodesics are Morse with the same gauge. We state the result just for the cases we shall use, finitely generated groups. Lemma 2.9. Let G be a group generated by the finite set S. If there exists a Morse gauge M so that for each n P N there is an M -Morse geodesic γ n in CaypG, Sq with diampγ n q ě 2n, then there exists a bi-infinite M -Morse geodesic γ : R Ñ CaypG, Sq.
Proof. Without loss of generality, we can assume that each γ n has the form γ n : r´n, ns Ñ CaypG, Sq with γp0q " e. Since there are only a finite number of paths of a specific length in CaypG, Sq, a diagonal argument produces a subsequence γ n k where γ n k | r´ℓ,ℓs " γ n ℓ | r´ℓ,ℓs whenever ℓ ď k. The map γ : R Ñ CaypG, Sq defined by γptq " γ n k ptq for |t| ă k is then an M -Morse geodesic, as the M -Morse γ n k exhaust γ.
2.3. Local quasi-geodesics and the local-to-global property. In this section, pX, dq will be a metric space and I will be a closed, but not necessarily bounded, interval in R.
Definition 2.10 (Local quasi-geodesic). Let L ą 0. The map γ : I Ñ X is an pL; λ, ǫq-local quasigeodesic if for any s ă t in I with |s´t| ď L, γ| rs,ts is a pλ, ǫq-quasi-geodesic. If, in addition, γ| rs,ts is M -Morse, then γ is an pL; M ; λ, ǫq-local Morse quasi-geodesic. We call the number L the scale of the local quasi-geodesic.
The following theorem of Gromov shows that hyperbolic spaces are characterized by the property that all local quasi-geodesics of sufficiently large scale are global quasi-geodesics.
The main topic of this paper are spaces where local Morse quasi-geodesics with sufficiently large scale are global Morse quasi-geodesics. In Sections 4 and 5, we show that this class of spaces includes CATp0q spaces, hierarchically hyperbolic spaces such as the mapping class groups and Teichmüller space, and spaces hyperbolic relative to spaces with the local-to-global property.  (1) One can equivalently define the Morse local-to-global property without reference to the map Φ as follows: X has the Morse local-to-global property if for every λ ě 1, ǫ ě 0 and Morse gauge M , there exist L, k, c, and N so that every pL; M ; λ, ǫq-local Morse quasi-geodesic in X is a global pN ; k, cq-quasi-geodesic. (2) The Morse local-to-global property applies to local Morse quasi-geodesics with both infinite and finite domains. However, when verifying a space has the Morse local-to-global property, it is sufficient to verify the property for only the local Morse quasi-geodesics with finite domains.
We now record a pair of basic lemmas about local quasi-geodesics the we will use throughout this paper. The first reduces the problem of showing a local quasi-geodesic is a global quasi-geodesic to showing that every subsegment of the local quasi-geodesic is uniformly close to a geodesic. Lemma 2.14. Let γ : I Ñ X be an pL; λ, ǫq-local quasi-geodesic and C ě 0. If L ą λp3C`ǫ`2q and for all rs, ts Ď I, γ| rs,ts is contained in the C-neighborhood of a geodesic from γpsq to γptq, then γ is a pλ 1 , ǫ 1 q-quasi-geodesic where λ 1 and ǫ 1 depend only on λ, ǫ, and C.
For the first inequality, let t 1 " s 0 ă s 1 ă¨¨¨ă s n " t 2 such that L{2 ď |s i´si`1 | ă L. Thus, each γ| rs i ,s i`1 s is a pλ, ǫq-quasi-geodesic and we have For the second inequality, let α be a geodesic connecting γpt 1 q and γpt 2 q. Since γ`rt 1 , t 2 s˘Ď N C pαq, there exists a map p : rt 1 , t 2 s Ñ impαq such that ppt 1 q " γpt 1 q, ppt 2 q " γpt 2 q, and d`ppsq, γpsq˘ď C for all s P pt 1 , t 2 q. Let γpt 1 q " x 0 , x 1 , . . . x n " γpt 2 q be a sequence of points of α so that 1 ď dpx i , x i`1 q ă 2 and d`γpt 1 q, γpt 2 q˘" n´1 ÿ Let I i be the closed subsegment of α between x i and x i`1 and define J i " p´1pI i q for each 0 ď i ď n´1. Since n ď 2d`γpt 1 q, γpt 2 q˘`1 and the desired inequality will follow if we can uniformly bound diampJ i q in terms of λ, ǫ, and C for each 0 ď i ď n´1. Suppose diampJ i q ą 0 for some 0 ď i ď n´1 and let s, r P J i with s ă r. By definition of J i and p we have By assumption, for all v P rs, rs, γpvq is within C of a geodesic from γpsq to γprq. This implies d`γpsq, γpvq˘ď 3C`2 for all v P rs, rs. (˚) Because L ą λp3C`ǫ`2q and γ is an pL; λ, ǫq-local quasi-geodesic, d`γpsq, γps`Lq˘ě L λ´ǫ ą 3C`2.
An important source of local quasi-geodesics are concatenations of geodesics. The next lemma shows that if a concatenation of Morse geodesics is a local quasi-geodesic, then is will automatically be a local Morse quasi-geodesic. Lemma 2.15. Let X be a geodesic metric space. For each i P t1,¨¨¨, nu, let γ i : ra i , a i`1 s Ñ X be an M -Morse geodesic with γ i pa i`1 q " γ i`1 pa i`1 q for 1 ď i ď n´1. If there is L ą 0 such that |a i`2´ai | ą L for each i and the concatenation γ " γ 1˚¨¨¨˚γn is an pL; λ, ǫq-local quasi-geodesic for some λ ě 1 and ǫ ě 0, then there is a Morse gauge M 1 depending on λ, ǫ, and M such that γ is an pL; M 1 ; λ, ǫq-local quasi-geodesic.
Since |a i`2´ai | ą L for each i, there are only three possibilities for the position of t 1 and t 2 in ra 1 , a n`1 s: (1) t 1 and t 2 both lie in ra j , a j`1 s for some j; (2) t 1 P ra j , a j`1 s and t 2 P ra j`1 , a j`2 s for some j; (3) t 1 P ra j , a j`1 s and t 2 P ra j`2 , a j`3 s for some j.
In all three cases, Lemma 2.7 implies any geodesic α connecting γpt 1 q and γpt 2 q is an N -Morse geodesic and that lies in the C-neighborhood of γ`rt 1 , t 2 s˘where C and N only depend on M . Lemma 2.8 thus provides a Morse gauge M 1 depending on λ, ǫ, and M such that γ| rt 1 ,t 2 s is M 1 -Morse. Therefore, γ is an pL; M 1 ; λ, ǫq-local quasi-geodesic.
2.4. Stable subgroups. Intimately connected with the study of Morse geodesics are stable subgroups of finitely generated groups. Stable subgroups are a strong generalization of quasiconvex subgroups of hyperbolic groups. The above definition of stable subgroup is equivalent to the definition originally given by Durham and Taylor in [DT15]. In Definition 2.16, the parameters pM, µq "measure" the stability of a subgroup H in the Cayley graph CaypG, Sq. While the specific pair pM, µq depends on the choice of finite generating set for G, the stability of the subgroup H does not. That is, if S and T are two finite generating sets for G and H is pM, µq-stable in CaypG, Sq, then H will be pM 1 , µ 1 q-stable in CaypG, T q for some M 1 and µ 1 depending on M , µ, S, and T .
Durham and Taylor generalized several properties of quasiconvex subgroups of hyperbolic groups to all stable subgroups. The key properties we need are summarized in the next proposition. Proposition 2.17 (Properties of Stable subgroups; [DT15]). Let G be a finitely generated group and H ă G a stable subgroup.
(2) H is a hyperbolic group.
(3) If K is a finitely generated, undistorted subgroup of H, then K is stable in G.
The third author and Antolín, Mj, Sisto, and Taylor independently studied the intersection properties of stable subgroups and showed that all infinite index stable subgroups have finite width and are finite index in their commensurators. ‚ H is finite index in the commensurator of H in G. ‚ H has finite width, i.e., there exist n so that if H is a set of at least n distinct cosets of H, then there exists g 1 H, g 2 H P H so that g 1 Hg´1 1 X g 2 Hg´1 2 is finite.
The most well studied examples of stable subgroups in the literature are stable cyclic subgroups. The generators of these subgroups are called Morse elements since the cyclic subgroup they generate will be a Morse quasi-geodesic in the Cayley graph. Definition 2.19 (Morse element). Let G be a finitely generated group with a finite generating set S and g P G. We say g is M -Morse with respect to S, if g has infinite order and xgy is pM, µq-stable in CaypG, Sq for some µ ě 0. A group element in G is Morse if it is M -Morse with respect to some finite generating set S.
While the specific Morse gauge of an element depends on the choice of generating set, whether or not an element is Morse is independent of choice of generating set.

Consequences of the Morse local-to-global property
We now give our main applications of the local-to-global property for Morse quasi-geodesics. In Section 3.1, we prove two combination theorems for stable subgroups of Morse local-to-global groups as well as several consequences of these theorems. In Section 3.2, we show that the algebraic translation length of conjugacy classes with a fixed Morse gauge is discrete. In Section 3.3, we prove our local condition for checking hyperbolicity.
3.1. Stable subgroup combination theorems. The primary results of this section are the following combination theorems for stable subgroups of Morse local-to-global groups. These results extend results of Gitik for quasiconvex subgroups of hyperbolic groups [Git99, Theorems 1, 2] and prove Theorems G and M from the introduction.
Theorem 3.1. Let G be a finitely generated group with the Morse local-to-global property and S be a fixed finite generating set for G. If P, Q are pM, µq-stable subgroups of G, then there exists C " CpM, µ, Sq ą 0 such that the following holds for all subgroups P 1 ď P and Q 1 ď Q.
(1) If P 1 X Q 1 " P X Q " I and I contains all elements of P 1 Y Q 1 whose word length in G is less than C, then xP 1 , Q 1 y -P 1˚I Q 1 . If P 1 and Q 1 are additionally finitely generated and undistorted in G, then the subgroup xP 1 , Q 1 y is stable in G.
(2) If P is malnormal in G, P 1 X Q " P X Q " I, and I contains all elements of P 1 whose word length in G is less than C, then xP 1 , Qy -P 1˚I Q. If P 1 is additionally finitely generated and undistorted in G, then the subgroup xP 1 , Qy is stable in G.
The proofs of (1) and (2) are quite similar to each other and to Gitik's original proofs in the case of hyperbolic groups. Before giving the proofs, we sketch the argument for the case where P 1 " P and Q 1 " Q.
The amalgamated product P˚I Q naturally surjects onto the subgroups xP, Qy. The goal is therefore to show this map is an isomorphism by proving any h P P˚I Q that is not in I " P X Q is non-trivial under the map P˚I Q Ñ G. The element h P P˚I Q´I can be decomposed as an alternating product p 1 q 1 . . . p n q n of elements of P and Q so that each p i and q i contains some subword that is not in P X Q. Since P and Q are stable subgroups, the path γ starting at the identity and labeled by p 1 q 1 . . . p n q n is a concatenation of Morse geodesics. By requiring that all short element of P and Q are contained in P X Q, we can mimic Gitik's proof in the hyperbolic case to show that γ is a local Morse quasi-geodesic with sufficiently large scale. We can then apply the Morse local-to-global property to show γ is actually a uniform Morse quasi-geodesic with small enough constants that γ cannot be a loop in the Cayley graph of G. Since γ is not a loop, p 1 q 1 . . . p n q n cannot be the identity and xP, Qy -P˚I Q. The stability of P and Q implies that the uniform Morse quasi-geodesic γ is contained in a regular neighborhood of xP, Qy, proving the subgroup is stable.
Proof of (1). There exist M and µ such that P and Q are pM, µq-stable in CaypG, Sq. Let δ " 4M p3, 0q and A be the number of elements of G with length less than 2µ`δ. Define ǫ " 4Aµ`δ and let M 1 be the Morse gauge so that a concatenation of M -Morse geodesics satisfying the hypothesis of Lemma 2.15 is an pL; M 1 ; 3, ǫq-local Morse geodesic whenever the concatenation is an pL; 3, ǫqlocal geodesic. Since CaypG, Sq has the Morse local-to-global property, there are L ą 0, k ě 1, c ě 0 and a Morse gauge N such that every pL; M 1 ; 3, ǫq-local Morse quasi-geodesic in CaypG, Sq is an pN ; k, cq-Morse quasi-geodesic. Let C " maxtL, kcu`1.
Form the abstract group P 1˚I Q 1 from isomorphic copies of P 1 and Q 1 . Let ι : P 1˚I Q 1 Ñ G be the natural map whose image is the subgroup xP 1 , Q 1 y ă G. We need to prove that the map ι is injective.
Consider an element h P P 1˚I Q 1 such that h R I. We will show that ιphq is not the identity in G. We can write h as a product h " p 1 q 1 p 2 q 2¨¨¨qm´1 p m where p i and q i satisfy the following: (1) q i P Q 1´I for 1 ď i ď m´1; (2) p 1 , p m P P 1 where p 1 (resp. p m q is a shortest representative of the coset p 1 I (resp. Ip m ); (3) For 2 ď i ď m´1, p i P P 1´I is a shortest representative of the double coset Ip i I. For 1 ď i ď m, let α i and β i be M -Morse geodesics in CaypG, Sq connecting the following points: ‚ α 1 connects e and p 1 ; ‚ α m connects p 1 q 1¨¨¨qm´1 and p 1 q 1¨¨¨qm´1 p m ; ‚ α i connects p 1 q 1¨¨¨qi´1 and p 1 q 1¨¨¨qi´1 p i for all 2 ď i ď m´1; ‚ β i connects p 1 q 1¨¨¨pi and p 1 q 1¨¨¨pi q i for 1 ď i ď m´1. Note, α 1 is degenerate if p 1 is trivial and α m is degenerate if p m is trivial, but all other α i and β i have length as least C ą L. Let γ " α 1˚β1˚α2˚β2˚¨¨¨˚βm´1˚αm . We claim that γ is an pL; M 1 ; 3, ǫq-local Morse quasi-geodesic.
Since each α i except α 1 and α m has length at least C ą L, Lemma 2.15 states it is sufficient to verify that γ is an pL; 3, ǫq-local quasi-geodesic. Let ψ be a subsegment of γ with parametrized length at most L. If ψ is contained entirely in a single α i or β i , then ψ is a p3, ǫq-quasi-geodesic. Otherwise, C ą L implies that ψ decomposes into two pieces η 1 and η 2 where, without loss of generality, η 1 Ď α i and η 2 Ď β i for some 1 ď i ď m´1. If η 3 is a geodesic in CaypG, Sq connecting the endpoints of ψ, then η 1 Y η 2 Y η 3 is δ-slim by Lemma 2.7.
In the proof of [Git99, Theorem 1], Gitik shows that if G is a δ-hyperbolic group, then ψ will be a p3, ǫq-quasi-geodesic. This argument only uses the fact that the triangle η 1 Y η 2 Y η 3 is uniformly δ-slim, that the segments α i and β i are contained in the µ-neighborhood of a coset of either P or Q respectively, that there are only A elements of G of length 2δ`µ, and the minimality of the choice of the p i . Since these facts remain true in this setting, we can apply the same argument to conclude that ψ is a p3, ǫq-quasi-geodesic. This implies γ is an pL; 3, cq-local quasi-geodesic and hence an pL; M 1 ; 3, ǫq-local Morse quasi-geodesic.
Since γ is an pL; M 1 ; 3, ǫq-local Morse quasi-geodesic, the Morse local-to-global property implies that γ is an pN ; k, cq-Morse quasi-geodesic. Now, the path γ connects the identity e and the element ιphq in CaypG, Sq. Since the parametrized length of γ is greater than C ě kc`1 and the distance between the endpoints of γ is positive, and ιphq is not the identity. This implies that ι is injective and xP 1 , Q 1 y -P 1˚I Q 1 .
If P 1 and Q 1 are finitely generated and undistorted in G, then they are both stable subgroups of G. To verify that xP 1 , Q 1 y is stable, it is sufficient to check that every element h P xP 1 , Q 1 y´I can be connected to the identity with a uniform quality Morse quasi-geodesic that is contained in a uniform neighborhood of xP 1 , Q 1 y. Since P 1 and Q 1 are stable, each of the α i and β i used to construct γ will be uniformly close to a coset of P 1 or Q 1 respectively. Thus, the pN ; k, cq-Morse quasi-geodesic γ will then be contained in a uniform neighborhood of xP 1 , Q 1 y, and xP 1 , Q 1 y will be a stable subgroup of G.
Proof of (2). There exist M and µ such that P and Q are pM, µq-stable in CaypG, Sq. Let δ " 4M p3, 0q and A be the number of elements of G with length less than 2µ`δ. Let Γ be the quotient of the action of P on CaypG, Sq. If v 0 P Γ is the vertex that represents the orbit of the identity, then let r be the number of vertices of Γ in the ball of radius µ`2δ centered at v 0 . Set ǫ " 4Aµ`δ`r 2`1 . Let M 1 be the Morse gauge so that a concatenation of M -Morse geodesics satisfying the hypotheses of Lemma 2.15 is an pL; M 1 ; 6, ǫq-local Morse geodesic whenever the concatenation is an pL; 6, ǫqlocal quasi-geodesic. Since CaypG, Sq has the Morse local-to-global property, there are L ą 0, k ě 1, c ě 0 and a Morse gauge N such that every pL; M 1 ; 6, ǫq-local Morse quasi-geodesic is an pN ; k, cq-Morse quasi-geodesic. Let C " maxtL, kcu`1.
Form the abstract group P 1˚I Q from isomorphic copies of P 1 and Q. As in the proof of (1), we need to show that the natural map of P 1˚I Q Ñ xP 1 , Qy ă G is injective. As before, for h P P 1˚I Q´I, we can pick a representation h " p 1 q 1 p 2 q 2¨¨¨qm´1 p m that satisfies the same properties as in (1) with the change that q i P Q´teu for each 1 ď i ď m´1. Continuing to follow the proof of (1), we select M -Morse geodesics α i and β i in CaypG, Sq, and let γ " α 1˚β1˚. . . α m´1˚βm´1˚αm . Once we show that γ is an pL; M 1 ; 6, cq-local Morse quasi-geodesic, the remainder of the proof will finish identically to the proof of (1).
As in (1), every α i except α 1 and α m will have length longer than C ą L. Thus, if γ is an pL; 6, ǫqlocal quasi-geodesic, then it will satisfy the hypotheses of Lemma 2.15 and be an pL; M 1 ; 6, ǫq-local Morse quasi-geodesic. Let ψ be a subpath of γ with parametrized length at most L. If ψ is contained entirely in a single α i or β i , then ψ is a p6, ǫq-quasi-geodesic. Otherwise, C ą L implies that ψ decomposes into three pieces η 1 , η 2 , η 3 . Without loss of generality, η 1 Ď α i , η 2 Ď β i and η 3 Ď α i`1 for some 1 ď i ď m´1 where at most one of η 1 or η 3 is empty. If η 4 is a geodesic in CaypG, Sq connecting the endpoints of ψ, then the rectangle In the proof of [Git99, Theorem 2], Gitik shows that if G is a δ-hyperbolic group, then ψ will be a p6, ǫq-quasi-geodesic. This argument only use the fact that P is malnormal, the rectangle η 1 Y η 2 Y η 3 Y η 4 is uniformly 2δ-slim, that the segments α i and β i are contained in the µ-neighborhood of a coset of either P or Q respectively, that G contains only A elements of length 2δ`µ, and the minimality of the choice of the p i . These facts all remain true in this setting, so we can apply the same argument to conclude that ψ is a p6, ǫq-quasi-geodesic.
The remainder of the proof now follows identically to the proof of (1).
Verifying that the intersection of two stable subgroups contains all the short elements of each subgroup can be quite challenging in practice, especially since the function Φ governing the Morse local-to-global property (and therefore the constant C) is often not explicit. However, one can circumvent this difficulty by utilizing the separability of subgroups.
Corollary 3.3. Let G be a finitely generated group with the Morse local-to-global property. Let P and Q be infinite, stable subgroups of G and I " P X Q.
(1) If I is separable and infinite index in both P and Q, then there exist infinite families of finite index subgroups P i ă P and If P is malnormal in G and I is separable and infinite index in P , then there exists an infinite family of finite index subgroups P i ă P with P i X Q " I and P i ă P i´1 so that each subgroup xP i , Qy is stable in G and xP i , Qy -P i˚I Q .
Proof. We give the proof for (1) and the proof for (2) is similar. Fix a finite generating S for G and let C be the constant from Theorem 3.1.
(1) for P and Q. For each of P and Q, there exists a finite set of elements outside of P X Q that have word length less than C. Since P X Q is separable in P and Q, there exist finite index subgroups P 1 ă P and Q 1 ă Q so that P 1 X Q 1 " P X Q and all elements of P 1 and Q 1 with word length less than C are contained in P 1 X Q 1 . Since P 1 and Q 1 are finite index in P and Q, they are finitely generated and undistorted in G and the conclusion follows (1). We produce the infinite family of subgroups by inductively separating P i and Q i from the short elements of P i´1 and Q i´1 that are not contained in P X Q.
Corollary 3.3 allows us to produce a plethora of examples where our combination theorem applies with a non-trivial intersection between the subgroups. We demonstrate an explicit example in rightangled Coxeter groups, which are Morse local-to-global by virtue of being cocompact CATp0q groups (or alternatively hierarchically hyperbolic groups).
Example 3.4. Let G be the right-angled Coxeter group defined by the graph Γ in Figure 1. Let Γ 1 be the subgraph of Γ with vertex set ta, b, c, d, mu and let Γ 2 be the subgraph of Γ with vertex set ta, f, e, d, p, nu. Let P and Q be the right-angled Coxeter subgroups of G defined by Γ 1 and Γ 2 respectively. Both P and Q are stable subgroups by [Tra19, Corollary 7.12], and P X Q is the virtually cyclic subgroup I generated by the vertex set ta, du. While the subgroups xP, Qy is isomorphic to P˚I Q, xP, Qy is not a stable subgroup by [Tra19, Corollary 7.12]. However, I is separable in both P and Q (see [Ago13, Corollary 1.2] for instance), so there are two finite index subgroups P 1 ă P and Q 1 ă Q such that P 1 X Q 1 " I, xP 1 , Q 1 y -P 1˚I Q 1 , and xP 1 , Q 1 y is a stable subgroup by Corollary 3.3. In addition to I being non-trivial, xP 1 , Q 1 y is a one-end group as it is virtually a graph of closed surface groups with cyclic edge groups [Wil12, Theorem 18].
When one of the subgroups is cyclic, we can separate the subgroup from short elements by raising the generator to sufficiently high powers. In Section 4.1, we will employ this idea to show that all Morse local-to-global groups that contain a Morse element are either virtually cyclic or contain a stable free subgroup of rank 2. Here, we employ this trick to generalize a theorem of Arzhantseva [Arz01, Theorem 1].
Corollary 3.5. Let G be a torsion free, Morse local-to-global group. If Q is a non-trivial infinite index stable subgroup of G, then there is an infinite order element h such that xQ, hy -Q˚xhy and xQ, hy is stable in G.
Proof. By Theorem 2.18, there exists g P G such that Q X gQg´1 " teu. Let k be a non-identity element of Q. Since Q is hyperbolic, xgkg´1y is undistorted in gQg´1 implying xgkg´1y is stable in G. Let P be the commensurator of xgkg´1y in G. By Theorem 2.18, xgkg´1y has finite index in P implying P is stable.
We now prove that P is malnormal. If u P G such that uP u´1 X P ‰ teu, then uP u´1 X P is infinite as G is torsion free. Since xgkg´1y is finite index in P , uxgkg´1yu´1 X xgkg´1y is finite index in uP u´1 X P . Therefore, uxgkg´1yu´1 X xgkg´1y is infinite and finite index in both cyclic subgroups uxgkg´1yu´1 and xgkg´1y. Thus, u is an element of P , the commensurator of xgkg´1y. This proves P is malnormal. By Theorem 3.1.(2) the subgroup generated by Q and h " gk n g´1 for n large enough is then stable in G and isomorphic to Q˚xhy.
Our final application of our combination theorem is to show that every normal subgroup of a Morse local-to-global group contains a Morse element. For the mapping class group of a surface more complex than a one-holed torus or a CATp0q group, the Morse elements are respectively the pseudo-Anosov and rank-1 elements. Thus, Corollary 3.6 shows every infinite normal subgroup of such a mapping class group or CATp0q group contains a pseudo-Anosov or rank-1 element respectively.
Corollary 3.6. Let G be a finitely generated group that contains an infinite order Morse element. If G has the Morse local-to-global property, then every infinite normal subgroup of G contains an infinite order Morse element of G.
Proof. Let h P G be an infinite order Morse element and N an infinite normal subgroup of G. Assume that no positive power of h is an element of N . The subgroup H " xhy is an infinite index stable subgroup of G because h is a Morse element. Since H is finite index in its commensurator in G (Theorem 2.18) and N is infinite, there exists an infinite number of distinct left cosets of H with representatives in N . Theorem 2.18 thus provides g P N such that H X gHg´1 " teu. By Theorem 3.1.(1), there is n ą 0 so that the elements h n and gh n g´1 generate a stable free subgroup. Since N is a normal subgroup and g P N , ph´ngh n qg´1 is an infinite order Morse element of N .
Remark 3.7. By [DMS10, Lemma 3.25], if the subgroup N is also finitely generated, then the Morse element of G it contains is also a Morse element of N with respect to the word metric on N .
3.2. Translation lengths for Morse elements. We now generalize a result stated by Gromov and proved by Delzant on the discreteness of the set of algebraic translation lengths of elements of hyperbolic groups. We show that in the case of Morse local-to-global groups, the same result applies to conjugacy classes with a fixed Morse gauge.
Definition 3.8. Let G be a group with finite generating set S. The algebraic translation length of g P G is defined to be τ G,S pgq " lim nÑ8 |g n | n .
The limit lim nÑ8 |g n | n always exists, because the function n Þ Ñ |g n | is sub-additive.
For a given finite generating set, the algebraic translation length depends only on the conjugacy class of an element. While being Morse is also a conjugacy class invariant, the specific Morse gauge of an element is not. Thus, we need to define a useful definition of Morse gauge for a conjugacy class of Morse elements. We do so by taking the Morse gauge for the elements in the conjugacy class with the shortest word length.
Definition 3.9 (Morse gauge of a conjugacy class). Let G be a finitely generated group with a finite generating set S. For each element g P G, let rgs denote the conjugacy class of g and shortprgsq be the collection of group elements in rgs with minimal length with respect to S. The conjugacy class rgs is M -Morse with respect to S if every element of shortprgsq is M -Morse.
The key use of the Morse local-to-global property in establishing discreteness of translation length is Lemma 3.11 below, which says M -Morse elements with minimal word length in their conjugacy class have a Morse quasi-axis with constants depending only on M .
Definition 3.10 (Quasi-axis). Let G be a finitely generated group with a finite generating set S. The quasi-axis for group element g P G and a geodesic α from e to g in CaypG, Sq is the path p g : p´8, 8q Ñ CaypG, Sq such that for all n P Z, p g restricted to the interval rpn´1q|g|, n|g|s is g n α.
Lemma 3.11. Let G be a finitely generated group with the Morse local-to-global property and S be a finite generating set for G. Let g P G be an infinite order element such that |g| is minimal in rgs. If g is M -Morse with respect to S, then any quasi-axis p g is an pN ; λ, ǫq-Morse quasi-geodesic in CaypG, Sq, where λ, ǫ, and N depend only on G, S, and M .
Proof. Let δ " 4M p3, 0q be the constant so that geodesic triangles with two M -Morse sides are δ-slim (Lemma 2.7). Let M 0 be the Morse gauge, depending only on M , so that any concatenation of M -Morse geodesics satisfying the hypothesis of Lemma 2.15 is an pL; M 0 ; 1, 4δ`4q-local Morse quasi-geodesic whenever the concatenation is an pL; 1, 4δ`4q-local quasi-geodesic. Let L, λ 0 , ǫ 0 be the constants and N 0 the Morse gauge such that every pL; M 0 ; 1, 4δ`4q-local Morse quasi-geodesic in CaypG, Sq is a global pN 0 ; λ 0 , ǫ 0 q-Morse quasi-geodesic in CaypG, Sq.
Let α be any geodesic from e to g and p " p g be the quasi-axis for g and α as described in Definition 3.10. We first use the Morse local-to-global property to show that if |g| ą L, then the quasi-axis p is an pN 0 ; λ 0 , ǫ 0 q-Morse quasi-geodesic.
Suppose |g| ą L and let t 1 , t 2 P p´8, 8q with t 1 ă t 2 and |t 2´t1 | ă L. Since the length of α is greater than L, p`rt 1 , t 2 s˘must be contained in either g n α or g n α Y g n`1 α for some n P Z. If p`rt 1 , t 2 s˘Ď g n α, then p| rt 1 ,t 2 s is an M -Morse geodesic, so suppose p`rt 1 , t 2 s˘Ę g n α for any n P Z. Without loss of generality, we can assume p`rt 1 , t 2 s˘Ď g´1α Y α.
Let s 1 , s 2 P rt 1 , t 2 s. Since p| rt 1 ,t 2 s is a concatenation of geodesics, the inequality d`pps 1 q, pps 2 q˘ď |s 2´s1 | holds by the triangle inequality. For the other inequality we can assume that pps 1 q P g´1α and pps 2 q P α.
Since p is an pL; 1, 4δ`4q-local quasi-geodesic, p satisfies the hypothesis of Lemma 2.15 as α has length more than L. Thus, p is an pL; M 0 ; 1, 4δ`4q-local Morse quasi-geodesic.
By the Morse local-to-global property of CaypG, Sq, Claim 1 implies that p " p g is an pN 0 ; λ 0 , ǫ 0 q-Morse quasi-geodesic when |g| ą L.
To finish the proof in the remaining cases, recall that L depends only on M , G, and S. Therefore, the number of M -Morse elements with word length at most L is bounded by a constant depending only on M , G, and S. This implies that there are constants λ 1 , ǫ 1 and a Morse gauge N 1 depending only on M , G, and S such that p g is an pN 1 ; λ 1 , ǫ 1 q-quasi-geodesic for each M -Morse element g satisfying |g| ď L. Thus, the lemma follows with λ " maxtλ 0 , λ 1 u, ǫ " maxtǫ 0 , ǫ 1 u, and N " maxtN 0 , N 1 u.
With Lemma 3.11 in hand, we can apply Delzant's argument to show discreteness of translation length for conjugacy classes with a fixed Morse gauge.
Theorem 3.12. Let G be a finitely generated group with the Morse local-to-global property. For all Morse gauges M and finite generating sets S, the set of algebraic translation lengths of elements of G whose conjugacy class is M -Morse with respect to S is a discrete subset of the rational numbers.
Proof. Let rgs be an M -Morse conjugacy class in G with respect to S and let u be the shortest representative of rgs. By Lemma 3.11, any quasi-axis p u is an pN ; λ, ǫq-Morse quasi-geodesic where N , λ, and ǫ depend only on M , G, and S. Let U`and U´be the endpoint of p u in the Morse boundary of CaypG, Sq (see [Cor17]   Definition 3.14. A geodesic metric space X is pR; δq-locally hyperbolic if for each point x P X and for each triple of points a, b, c P Ball R pxq, any geodesic triangle with vertices a, b, and c is δ-slim.
To prove Theorem 3.15, we show local hyperbolicity implies every geodesic is uniformly locally Morse. We can then apply the Morse local-to-global property to conclude that every geodesic is Morse. This implies hyperbolicity by Lemma 2.7. We first record an auxiliary lemma.
Lemma 3.16. For each δ ą 0 there is a Morse gauge M so that the following holds. Let X be an pR; δq-locally hyperbolic space and x P X. Let α be a geodesic in X and β be a pλ, ǫq-quasi-geodesic with the same endpoints as α. If α and β are both contained Ball R{4 pxq, then the Hausdorff distance between β and α is at most M pλ, ǫq.
Proof. Any geodesic connecting two points in α Y β is contained in Ball R pxq. Since X is pR; δqlocally hyperbolic, we can follow the proof of [BH99, Theorem III.H.1.7] to find a Morse gauge for α depending only on δ.
Proof of Theorem 3.15. Let δ ą 0 and M be the Morse gauge, depending only on δ, from Lemma 3.16. By increasing M , we can assume that M pλ, ǫq ě λ 2 pλ`2ǫq`ǫ`1 for all λ ě 1 and ǫ ě 0. Let N be the Morse gauge and L, k, c be the constants such that every pL; M ; 1, 0q-local Morse quasi-geodesic in X is a global pN ; k, cq-Morse quasi-geodesic, i.e., pL, N, k, cq " ΦpM, 1, 0q. Let R " 8L 3`4 L`1 and assume that X is pR; δq-locally hyperbolic. If we show that every geodesic of X is an pL; M ; 1, 0q-Morse quasi-geodesic, then every geodesic of X is N -Morse. By Lemma 2.7, this implies that X is δ 1 -hyperbolic where δ 1 " 4N p3, 0q.
Thus, α and the subsegment of γ between αpaq and αpbq both lie in the ball Ball R{4`α paq˘. The Hausdorff distance between α and the geodesic segment of γ between αpaq and αpbq is then at most M pλ, ǫq by Lemma 3.16. This verifies that the geodesic γ is an pL; M ; 1, 0q-local Morse quasi-geodesic. Since X has the Morse local-to-global, γ is N -Morse. Hence, X is δ 1 -hyperbolic where δ 1 " δ 1 pδ, Φq.

Spaces with the Morse local-to-global property
In this section, we prove unconstricted spaces, CATp0q spaces, and hierarchically hyperbolic spaces with a technical condition have the Morse local-to-global property. This covers all of the examples given in Theorem D, except the fundamental groups of closed 3-manifolds with Nil or Sol components in their prime decomposition. The proof for all closed 3-manifold groups is contained in the next section on relative hyperbolicity. In section 4.1, we also give examples of groups that contain infinite Morse quasi-geodesics, but do not have the Morse local-to-global property. As we will not be needing the full breath of the definitions, we opt to give the salient properties of each class of spaces that we need and forgo giving the definitions in full. We direct the reader to the following sources for detailed background on the spaces in question: unconstricted spaces [DS05], CATp0q spaces [BH99], and hierarchically hyperbolic spaces [BHS19].  Since Morse quasi-geodesics are preserved by quasi-isometries, being Morse limited is a quasiisometry invariant.
When a space is Morse limited, it trivially satisfies the Morse local-to-global property by choosing the local scale to be larger than the bound on the length of the domain of a Morse quasi-geodesic.
Lemma 4.2. Let X be a metric space. If X is Morse limited, then X has the Morse local-to-global property.
Proof. Let B ě 0 so that the domain of any pM ; λ, ǫq-Morse quasi-geodesic has diameter at most B. Let L ą B and suppose γ : I Ñ X is an pL; M ; λ, ǫq-local Morse quasi-geodesic. If diampIq ą B, then there exists ra, bs Ď I with B ă |a´b| ď L. However, this is a contradiction as γ| ra,bs would then be an pM ; λ, ǫq-Morse quasi-geodesic with domain longer than B. Thus, diampIq ď B ă L and γ is an pM ; λ, ǫq-Morse quasi-geodesic.
While Lemma 4.2 is a simple observation, it plays a key role when combined with the relative hyperbolicity results of Section 5. In particular, the fundamental groups of all closed 3-manifolds will have the Morse local-to-global property as they are hyperbolic relative to subgroups that have the Morse local-to-global property either by being hierarchically hyperbolic space or by being Morse limited.
A  Theorem 4.4 ([DS05, Theorem 6.5, Corollary 6.14]). Let G be finitely generated group that is not virtually cyclic. If G either satisfies a law or has a central, infinite cyclic subgroup, then G is unconstricted.
Examples of groups that are not unconstricted included those that contain a bi-infinite Morse geodesics. Proposition 4.6. If a finitely generated group G is unconstricted, then G is Morse limited.
Proof. We prove the contrapositive Let S be a finite generating set for G and suppose there is a sequence, γ n : I n Ñ CaypG, Sq, of pM ; λ, ǫq-Morse quasi-geodesics with diampI n q Ñ 8 as n Ñ 8. By Lemma 2.8, there exists a Morse gauge M 1 so that CaypG, Sq contains M 1 -Morse geodesics of arbitrarily long length. This implies G is not unconstricted as CaypG, Sq will contain an infinite M 1 -Morse geodesic by applying Lemma 2.9 Morse limited groups give a structural dichotomy to Morse local-to-global group: either the group is Morse limited or it contains a Morse element.
Theorem 4.7. Let G be a finitely generated group with Morse local-to-global property. Either G contains a Morse element or G is Morse limited.
Since Lemma 2.9 guarantees that a group containing arbitrarily long Morse geodesics must contain an infinite Morse geodesic, Theorem 4.7 follows immediately from the next proposition plus Lemma 2.9.
Proposition 4.8. Let G be a finitely generated group with the Morse local-to-global property. If there is some finite generating set S of G such that the Cayley graph CaypG, Sq contains a Morse geodesic ray, then G contains a Morse element.
For each integer i ě 0, let α i be the subsegment of α from αpi¨2Lq to α`pi`1q¨2L˘. There are two non-negative integers m and n such that n´m ě 2 and α m and α n are both labeled by same word w 1 in S. Let w 2 be the word in S labeling the subsegment of α from the end of α m to the beginning of α n . By construction, |w 1 |, |w 2 | ě 2L and any geodesic labeled by any subword of w 1 w 2 or w 2 w 1 is an M -Morse geodesic.
Let ℓ " |w 1 |`|w 2 | and define β : p´8, 8q Ñ CaypG, Sq to be the path such that βp0q " e and for each i P Z, β| riℓ,pi`1qℓs is labeled by the word w 1 w 2 . Then, β is an pL; M ; 1, 0q-local Morse quasi-geodesic and hence an pN ; λ, ǫq-Morse quasi-geodesic. This implies that w 1 w 2 represent a Morse element of G.
By applying our combination theorem for stable subgroups (Theorem 3.1), we expand Theorem 4.7 to show that Morse local-to-global groups that are not Morse limited are either virtually cyclic or contain a stable free subgroups of rank 2.
Corollary 4.9. If G is a finitely generated group with the Morse local-to-global property, then G satisfies exactly one of the following: (1) G is Morse limited; (2) G is virtually infinite cyclic; (3) G contains a stable, free subgroup of rank 2.
Proof. Assume that G is not Morse limited and not virtually a cyclic group. By Theorem 4.7, G must contain a Morse element h. The subgroup H " xhy is then an infinite index stable subgroup of G. Thus, there is an element g P G such that gHg´1 XH " teu (Theorem 2.18). By Theorem 3.1, there is n ą 0 such that the elements h n and gh n g´1 generate a stable free subgroup of rank 2.  (1) In [Fin], Finks the author gives an example of a finitely generated, torsion group that contains a bi-infinite Morse geodesic. Theorem 4.7 shows this groups is not Morse local-toglobal as it is not Morse limited and does not contain a Morse element. (2) In [OsOS09, Theorem 1.12], Osin, Ol'shanskii, and Sapir produce a finitely generated, nonvirtually cyclic group G where every proper, non-trivial subgroup is infinite cyclic and stable. Every non-identity element of G is Morse, but G does not contains a free subgroup of rank 2. Thus, G is not Morse local-to-global by Corollary 4.9.
Since the direct product of any group with Z is unconstricted and hence Morse limited, the above examples show the Morse local-to-global property does not descend to undistorted finitely generated subgroups. 4.2. CATp0q spaces. The main tool we need to establish the Morse local-to-global property in CATp0q spaces is the closest point projection onto geodesics. For the remainder of this section π γ will denote the closest point projection onto the geodesic γ described in the next lemma. (1) There exists a continuous function π γ : X Ñ γ so that for all x P X, π γ pxq is the unique point in γ that minimizes the distance from x to γ.
We also need a characterization of Morse geodesics in CATp0q spaces in term of the closest point projection established by Charney and Sultan.
Definition 4.12. Let D ě 0. A geodesic γ in a CATp0q space X is D-contracting if for all x, y P X with dpx, yq ă dpx, γq, the distance d`π γ pxq, π γ pyq˘is at most D. Finally, we need a lemma of the third author about concatenating geodesics in any geodesic metric space.
We prove the Morse local-to-global property. First, we use Lemma 4.11.(2) to show that a local Morse quasi-geodesic of sufficient scale is close to the geodesic between its end points (Proposition 4.15). This implies the local Morse quasi-geodesic is a global quasi-geodesic by Lemma 2.14. Next, we use the contracting characterization of Morse geodesics to prove this global quasi-geodesic is Morse (Theorem 4.16).
Proposition 4.15. Let X be a CATp0q space. For each λ ě 1, ǫ ě 0, and Morse gauge M , there are constants ℓ ě 0 and C ě 0 such that the following holds. If L ě ℓ and γ : ra, bs Ñ X is an pL; M ; λ, ǫq-local Morse quasi-geodesic and β is the geodesic connecting γpaq and γpbq, then the Hausdorff distance between γ and β is less than C.
Proof. For any two points x, y P X, let rx, ys denote the unique geodesic from x to y in the CATp0q metric on X. Let M 1 be the Morse gauge provided by Lemma 2.8 so that any geodesic connecting the endpoints of an pM ; λ, ǫq-Morse quasi-geodesic is M 1 -Morse. Let K 1 " M p1, 0q, K 2 " M 1 p5, 0q, and R " 2pK 2`K1`1 q. Let L ě ℓ " 4λp44R`ǫ`K 1 q.
Let γ : ra, bs Ñ X be an pL; M ; λ, ǫq-local Morse quasi-geodesic and β be the geodesic connecting γpaq and γpbq. We first show that γ Ď N C 1 pβq for some C 1 depending only on M , λ, and ǫ.
Proof. Assume for the purposes of contradiction that dpz, βq ě 2R. We shall create a contradiction to Lemma 4.11.(2) by showing there is w 1 P rx, zs and w 2 P rz, ys such that the distances dpw 1 , βq and dpw 2 , βq are both strictly less than the distance dpz, βq. We only give the proof for the existence of w 1 as the proof for the existence of w 2 is entirely analogous.
We now prove that β also lies in the C 2 -neighborhood of γ where C 2 depends only on λ, ǫ, and M . Let a " t 0 ă t 1 ă¨¨¨ă t n " b with |t i`1´ti | ă ℓ. Since each γ| rt i ,t i`1 s is a pλ, ǫq-quasi-geodesic, the distance between γpt i q and γpt i`1 q is bounded above by λℓ`ǫ. Let u 0 " γpaq, u n " γpbq, and for each 1 ď i ď n´1, let u i be a point in β such that the distance between γpt i q and u i is at most C 1 . By the triangle inequality, the distance between u i and u i`1 is at most λℓ`ǫ`2C 1 . If u P β, then u must lie in a subsegment of β with endpoints u i and u i`1 for some 0 ď i ă n. Therefore, the distance between u and γpt i q is at most ℓλ`ǫ`3C 1 . This implies that β lies in the pλℓ`ǫ`3C 1 q-neighborhood of γ.
Proof. Let ℓ and C be the constants from Proposition 4.15 for λ, ǫ, and M . Let L ě λp3C`ǫ2`ℓ q`1 and let γ : ra, bs Ñ X be an pL; M ; λ, ǫq-local Morse quasi-geodesic in a CATp0q space X. By Proposition 4.15, every subsegment of γ in contained in the C-neighborhood of the geodesic between its endpoints. Since L ą λp3C`ǫ`2q, Lemma 2.14 implies γ is a pλ 1 , ǫ 1 q-quasi-geodesic for λ 1 and ǫ 1 depending ultimately only on λ, ǫ, and M .
To show γ is Morse, consider the geodesic β connecting γpaq and γpbq. By Theorem 4.13 and Lemma 2.8, it suffices to show that β is D-contracting for some D depending only on M , λ, and ǫ. Let M 1 be the Morse gauge so that any geodesic with endpoints in the C-neighborhood of an pM ; λ, ǫq-Morse quasi-geodesic is M 1 -Morse. Let D ą 0 so that any M 1 -Morse geodesic in X is D-contracting and suppose L ą λpD`2C`ǫ`1q. Note, such a D depends ultimately only on M , λ, and ǫ.

4.3.
Hierarchically hyperbolic spaces and Morse detectability. For hierarchically hyperbolic spaces, the Morse local-to-global property follows from a result of Abbott, Behrstock, and Durham that established most hierarchically hyperbolic spaces admit a projection onto a hyperbolic space that detects when a quasi-geodesic is Morse. We prove that any space with this Morse detectability property has the Morse local-to-global property. In addition to simplifying the proof in the case of hierarchically hyperbolic spaces, this approach also provides an avenue for producing new examples of Morse local-to-global spaces.
Definition 4.17. A metric space X is Morse detectable if there exists a δ-hyperbolic space Y and a coarsely Lipschitz map π : X Ñ Y such that for every pλ, ǫq-quasi-geodesic γ : ra, bs Ñ X, the following holds.
Theorem 4.18. If X is a Morse detectable metric space, then X has the Morse local-to-global property.
Proof. Let Y be the δ-hyperbolic space and π : X Ñ Y be the coarsely Lipschitz map satisfying Definition 4.17. Fix λ ě 1, ǫ ě 0, and Morse gauge M . Since X is Morse detectable, there exists k ě 1 and c ě 0 depending on λ, ǫ, δ, and M so that if η is an pM ; λ, ǫq-Morse quasi-geodesic, then π˝η is a pk, cq-quasi-geodesic. By the local-toglobal property for quasi-geodesics in hyperbolic spaces (Theorem 2.11), there are constants L ą 2ǫ, k 1 ě 1 and c 1 ě 0 depending only on k, c, and δ such that any pL; k, cq-local quasi-geodesic in Y is a pk 1 , c 1 q-quasi-geodesic.
Abbott, Behrstock, and Durham showed that all hierarchically hyperbolic spaces satisfying a minor technical condition are Morse detectable [ABD21]. This covers all of the natural examples of hierarchically hyperbolic spaces as outlined in Corollary 4.21. Abbott, Behrstock, and Durham provide an explicit description of the space and projection map for Morse detectability. In the case of the mapping class group, the space is the curve graph and the projection map is the subsurface projection of Masur and Minsky.  Question 4.22. Is OutpF n q or Outer Space Morse detectable for n ě 3?

Spaces hyperbolic relative to Morse local-to-global spaces
In this final section, we show the Morse local-to-global property is inherited under relative hyperbolicity.
Theorem 5.1. Let X be a geodesic metric space that is hyperbolic relative to a collection of peripheral subsets P. If each element of P has the Φ-Morse local-to-global property, then X has the Ψ-Morse local-to-global property.
If a group G is hyperbolic relative to subgroups H 1 , . . . , H n , then the Cayley graph of G with respect to any finite generating set is hyperbolic relative to the collection of left cosets of the H i . Since each coset of the H i is isometric to H i , if each H i has the Φ i -Morse local-to-global property, then there exists Φ so that every coset of one of the H i has the Φ-Morse local-to-global property. Theorem 5.1 then implies G is also Morse local-to-global, proving Theorem E from the introduction.
Theorem 5.1 implies that the Morse local-to-global property is closed under free products of finitely generated groups. Combining this with the work in Section 4, we deduce that the fundamental groups of all close 3-manifolds have the Morse local-to-global property.
Corollary 5.2. If M is a closed 3-manifold, then π 1 pM q has the Morse local-to-global property.
Proof. By the geometrization of closed 3-manifolds, if M is a closed 3-manifold, then M has a prime decomposition M " M 1 #M 2 # . . . #M n where each M i is either geometric or has a mixed geometry. Thus, π 1 pM qπ 1 pM 1 q˚¨¨¨˚π 1 pM n q where each π 1 pM i q is either virtually solvable or a hierarchically hyperbolic space with the bounded domain dichotomy [BHS19, Theorem 10.1]. In both cases, π i pM q has the Morse local-to-global property by Section 4.1 or Corollary 4.21. Since π 1 pM q is hyperbolic relative to the collection of left cosets of π 1 pM 1 q, . . . , π 1 pM n q, Theorem 5.1 implies π 1 pM q has the Morse local-to-global property.
The proof of Theorem 5.1 is considerably longer and more technical than our previous proofs that spaces have the Morse local-to-global property. To guide the reader, we give an outline of the proof in Section 5.2 after collecting some required facts about relatively hyperbolic spaces in Section 5.1. The proof of Theorem 5.1 then spans Sections 5.3 to 5.6. As part of the proof, we investigate features of local quasi-geodesics in relatively hyperbolic spaces that maybe of independent interest. These include showing that local quasi-geodesics that "avoid" all peripheral subsets are actually global Morse quasi-geodesics (Section 5.3) and developing a notion of "deep points" for a local quasi-geodesic that decomposes a local quasi-geodesic into pieces that alternate between avoiding and passing through peripheral subsets (Section 5.4).

5.1.
Background on relatively hyperbolic spaces. If P is a collection of subsets of a geodesic metric space X, let p X be the space obtained from X by adding a point c P for each P P P and connecting c P to every element of P by an edge of length 1. The space X is hyperbolic relative to P if p X is δ-hyperbolic for some δ ě 0 and X satisfies a bounded subset penetration property that controls how geodesics travel through the elements of P (see [Sis] for a complete description and several equivalent definition of a relatively hyperbolic space).
For the remainder of this section, X will be a fixed geodesic metric space that is hyperbolic relative to a collection of peripheral subsets P, δ will be the constant so that p X is δ-hyperbolic, and π : X Ñ p X will denote the distance non-increasing inclusion of X into p X. We now recall several facts about the relatively hyperbolic space X. The first says that the peripherals must be isolated away from each other. There is increasing function F : r0, 8q Ñ r0, 8q so that for all P, U P P we have: A key tool in our study of relatively hyperbolic spaces is the coarse closest point projection onto peripheral subsets, denoted by π P : X Ñ P for all P P P. The basic properties of π P are outlined in the following lemma.
Lemma 5.4 (Properties of projection onto peripherals; [Sis13]). There exist µ ě 0 so that for each P P P there is a pµ, µq-coarsely Lipschitz map π P : X Ñ P such that the following hold for all P, U P P.
(4) For all x P X, if γ is a geodesic in X from x to π P pxq, then diampπ P pγqq ď µ.
The next two lemmas will frequently be used in tandem. Together they imply that if two points x, y P X project significantly far apart on a peripheral P , then every quasi-geodesic connecting x and y must travel close to P for a distance comparable to the distance between π P pxq and π P pyq. Lemma 5.5 (Linear quasiconvexity of peripherals; [DS05,Lemma 4.5]). For every λ ě 1 and ǫ ě 0 there exists r " rpλ, ǫq ě 1 such that for every R ě 1 if γ is a pλ, ǫq-quasi-geodesic joining points in N R pP q, then γ Ď N rR pP q.
The projections onto peripheral subsets along with the map π : X Ñ p X produce a distance formula where distances in X can be approximated by summing distances in the projections.
Theorem 5.7 (The distance formula; [Sis13, Theorem 3.1][BHS19, Theorem 6.10]). There exists T 0 ě 0 so that for all T ě T 0 , there is A ě 1 so that for all x, y P X we have The most important consequence of the distance formula for our proof of Theorem 5.1 are that quasi-geodesics whose projection to each peripheral subset is uniformly bounded will quasiisometrically embed into p X and be Morse quasi-geodesics of X.
(1) For every C ą 0, there exists k ě 1 and c ě 0 so that if diam`π P pγq˘ď C for all P P P, then π˝γ is a pk, cq-quasi-geodesic in p X.
(2) For every k ě 1 and c ě 0, there exists C ą 0 so that if π˝γ is a pk, cq-quasi-geodesic in p X, then diam`π P pγq˘ď C for all P P P.
Proof. Item (1) is a direct consequence of the distance formula. For Item (2), assume π˝γ is a pk, cq-quasi-geodesic and let Q ą 0 and R " Rpλ, ǫq be the constants from Lemma 5.6. If diampπ P pγqq ą λrQ`2R`2`ǫ`kp2`2Rq`c`1qs for some P P P, then there is t, s in the domain of γ so that γptq, γpsq P N R pP q and |t´s| ą kp2`2Rq`c. However, this would be a contradiction since |t´s| ď kp2`2Rq`c as π˝γptq, π˝γpsq are both within R`1 of the cone point c P and π˝γ is a pk, cq-quasi-geodesic. Thus, we have diampπ P pγqq ď λrQ`2R`2`ǫ`kp2`2Rq`c`1qs for all P P P.
Corollary 5.9 (Quasi-geodesics with bounded projections are Morse). For each C ě 0, λ ě 1, and ǫ ě 0, there exists a Morse gauge M so that if γ is a pλ, ǫq-quasi-geodesic in X and diampπ P pγqq ď C for all P P P, then γ is M -Morse.
Since diam`π P pαq˘ď C 1 and diam`π P pγq˘ď C, π˝α and π˝γ are respectively pk 1 , c 1 q and pλ 1 , ǫ 1 q-quasi-geodesic in p X where pk 1 , c 1 q and pλ 1 , ǫ 1 q depend only on k, c, C and λ, ǫ, C respectively (Corollary 5.8). Since p X is δ-hyperbolic, there exists a Morse gauge N , depending on λ 1 , ǫ 1 , and δ, so that π˝γ is N -Morse and π˝α is contained in the N pk 1 , c 1 q-neighborhood of π˝γ. For x P α, let y P γ so that d p X px, yq ď N pk 1 , c 1 q. Since d P px, yq ď C`C 1 for all P P P, the distance formula (Theorem 5.7) produces C 2 " C 2 pλ, ǫ, C, k, cq so that d X px, yq ď C 2 . Thus, γ is M -Morse where M depends on λ, ǫ, C, and δ.
Our last preliminary result is an adaption of the work of Hruska [Hru10,Proposition 8.14] and Sisto [Sis,Proposition 5.7] to fit our needs. In the sequel, we say a map γ : ra, bs Ñ X is an unparametrized pλ, ǫq-quasi-geodesic if there exists a homeomorphism f : ra 1 , b 1 s Ñ ra, bs so that γ˝f is a pλ, ǫq-quasi-geodesic of X.
Proof. It is sufficient to verify that π˝γ is within finite Hausdorff distance of the image of a geodesic in p X. Since X is a geodesic space, we can assume γ is a continuous quasi-geodesic without any loss of generality [BH99, Lemma III.H.1.11].
By [Sis13, Lemma 1.14], there is a continuous pλ 1 , ǫ 1 q-quasi-geodesic η : ra 1 , b 1 s Ñ X so that π˝η is an unparametrized geodesic in p X. By [Sis, Proposition 5.7], there is a constant D " Dpλ, ǫq and sequences a " q 0 ď p 1 ă q 1 ă¨¨¨ă p n ă q n ď p n`1 " b and a 1 " q 1 0 ď p 1 1 ă q 1 1 ă¨¨¨ă p 1 n ă q 1 n ď p 1 n`1 " b 1 so that ‚ the Hausdorff distance between γ| rq i ,p i`1 s and η| rq 1 i`1 s is bounded by D for i P t0, . . . , nu; ‚ γ| rp i`1 ,q i`1 s , and η| rp 1 i`1 ,q 1 i`1 s are both contained in the D-neighborhood of the same peripheral P i P P for i P t0, . . . , n´1u. This implies the Hausdorff distance between π˝γ| rq i ,p i`1 s and π˝η| rq 1 i ,p 1 i`1 s is also at most D for i P t0, . . . , nu and diam`πpγ| rp i`1 ,q i`1 s q˘ď 2D`2 for all i P t0, . . . , n´1u. Thus, the Hausdorff distance between π˝γ and the unparametrized geodesic π˝η is bounded by a constant depending only on λ and ǫ as desired.
Before continuing, we set constants that we shall use for the remainder of the section. We fix a pair of quasi-geodesic constants λ ě 1 and ǫ ě 0 and then fix the following: ‚ µ the constant from Lemma 5.4; ‚ F : r0, 8q Ñ r0, 8q the function from Lemma 5.3; ‚ r " rpλ, ǫq the constant from Lemma 5.5; ‚ Q and R " Rpλ, ǫq the constants from Lemma 5.6. With these constants fixed, we define θ " θpλ, ǫq to be θ " 100λµpλ`ǫ`µ`F p2rRq`r`R`rR`Q`1q.
Henceforth, whenever λ, ǫ, µ, r, Q, R, or θ are written, they implicitly refer to the constants outlined above. Note, r, R, and θ depend on λ and ǫ, while µ, F , and Q do not. Further, I will always denote a compact interval of R for the remainder of the section.

5.2.
A sketch of the proof of Theorem 5.1. Similar to the case of CATp0q spaces, the key to proving Theorem 5.1 is to establish that, when every peripheral subset has the Φ-Morse local-toglobal property, every local Morse quasi-geodesic of sufficient scale in X is finite Hausdorff distance from any quasi-geodesic between its endpoints. However, unlike the CATp0q case, the argument requires substantially more care and technicalities. The main steps of the proof are outlined below with work on the proof beginning in the next section.
Step 1: Use the local-to-global property of quasi-geodesics in the hyperbolic space p X to prove a local version of Corollary 5.9, i.e., any local quasi-geodesic where the projection of every local segment is uniformly bounded is a Morse quasi-geodesic (Section 5.3).
Step 2: Show any local Morse quasi-geodesic γ has a decomposition γ " σ 0˚α1˚σ1˚¨¨¨˚αn˚σn and a collection of peripheral subsets tP 1 , . . . , P n u so that ‚ each α i is sufficiently long and contained in the rR-neighborhood of the peripheral subset P i ; ‚ the projection of each local segment of each σ i to every peripheral subset is uniformly bounded (Section 5.4).
Step 3: Argue each σ i and α i are Morse quasi-geodesics. For the σ i , this follows from Step 1 since projection of each local segment of each σ i to every peripheral subset is uniformly bounded. For the α i , we apply the Morse local-to-global property of each of the peripherals P i whose rR-neighborhood contains α i (Corollary 5.23).
Step 4: Prove the projection of the endpoints of γ to each P i are coarsely equal to the projections of the endpoints of α i to P i . We show this by establishing P 1 , . . . , P n are "linearly ordered" by γ, that is, if 1 ď i ă j ă k ď n, the projection of P i onto P k is coarsely equal to the projection of P j onto P k (Section 5.5).
Step 5: Since the α i are sufficiently long, Step 4 ensures that the projection of the endpoints of γ to each of the P i are far enough apart that we can apply Lemma 5.6 to concluded that every quasi-geodesic between the endpoints of γ must pass close to the endpoints of each α i (Lemma 5.33).
Step 6: Since each of the σ i and α i are Morse quasi-geodesics, Step 5 ensures any quasi-geodesic between the endpoints of γ will be within finite Hausdorff distance from γ. This directly implies γ has the Morse property and Lemma 2.14 will imply that γ is a global quasigeodesic.

5.3.
Local quasi-geodesics with bounded projections. We now study local quasi-geodesics where the projection of local subsegments to all peripheral subsets are uniformly bounded.
Definition 5.11. An pL; λ, ǫq-local quasi-geodesic γ has D-bounded projections if every subsegment σ Ď γ of parametrized length at most L has diampπ P pσqq ď D for all P P P.
The goal is to prove a local version of Corollary 5.9. That is, for each D there exists a local scale L so that pL; λ, ǫq-local quasi-geodesics with D-bounded projections are global Morse quasi-geodesics.
To prove Corollary 5.12, we first show that local quasi-geodesics with bounded projections are quasi-geodesics in p X .
Proof. Let k and c be so that every pλ, ǫq-quasi-geodesic in X is a unparametrized pk, cq-quasigeodesic in p X (Lemma 5.10). Let ℓ, λ 1 , and ǫ 1 be such that every pℓ; k 3 , 5k 2 cq-local quasi-geodesic in the hyperbolic space p X is a global pλ 1 , ǫ 1 q-quasi-geodesic in p X (Theorem 2.11). Let A ě 1 be the constant from the distance formula (Theorem 5.7) with T " T 0`D . Assume L 1 ě 20ℓA 2 λkp1`ǫ`cq and let γ : I Ñ X be an pL 1 ; λ, ǫq-local quasi-geodesic with D-bounded projections. We assume diampIq ą L 1 as the proposition follows from Lemma 5.10 and the distance formula when diampIq ď L 1 .
For the first item, let a 0 ă a 1 ă¨¨¨ă a n be elements of I such that ra 0 , a n s " I and L 1 {4 ď |a i´ai`1 | ď L 1 {2 for all 0 ď i ď n´1. Since π˝γ| ra i ,a i`1 s is an unparametrized pk, cq-quasigeodesic for each 0 ď i ď n´1, there exists real numbers b 0 ă b 1 ă¨¨¨ă b n and a homeomorphism f : rb 0 , b n s Ñ ra 0 , a n s such that f pb i q " a i for 0 ď i ď n and π˝γ˝f | rb i ,b i`1 s is a pk, cq-quasi-geodesic in p X for each 0 ď i ď n´1. Let r γ " π˝γ˝f . The next two claims show r γ is an pℓ, k 3 , 5k 2 cq-local quasi-geodesic in p X. This implies r γ is a pλ 1 , ǫ 1 q-quasi-geodesic of p X by the local-to-global property of quasi-geodesics in the hyperbolic space p X.
Proof of Claim 3. Fix i P t0, 1, . . . , nu. Since γ has D-bounded projections, the distance formula with threshold T " T 0`D provides d X pγpa i q, γpa i`1 qq ď A¨d p X`p γpa i q, p γpa i`1 q˘`A. We show 2ℓ ď |b i´bi`1 | with the following calculation.
Proof of Claim 4. Let s, t P rb 0 , b n s with |s´t| ď ℓ. By Claim 3, we can assume there exist i P t1, . . . , n´1u so that s P rb i´1 , b i s and t P rb i , b i`1 s. It suffices to show that r γ| rb i´1 ,b i`1 s is a pk 3 , 5k 2 cq-quasi-geodesic. Now, r γ| rb i´1 ,b i`1 s is a concatenation of the pk, cq-quasi-geodesics r γ| rb i´1 ,b i s and r γ| rb i ,b i`1 s . Further, r γ| rb i´1 ,b i`1 s is a unparametrized pk, cq-quasi-geodesic because r γ| rb i´1 ,b i`1 s is a reparametrization of p γ| ra i´1 ,a i`1 s and p γ| ra i´1 ,a i`1 s is an unparametrized pk, cq-quasi-geodesic since |a i´1´ai`1 | ď L 1 . Thus, Lemma 5.14 below implies r γ| rb i´1 ,b i`1 s is a pk 3 , 5k 2 cq-quasi-geodesic.
For the second item, let t 1 , t 2 P I with |t 1´t2 | ď L 1 and let γpt 1 q " x 1 , γpt 2 q " x 2 . Using the distance formula as in Claim 3, we get Since π : X Ñ p X is distance non-increasing, we also have d p X pπpx 1 q, πpx 2 qq ď d X px 1 , x 2 q ď λ|t 1´t2 |`ǫ. These calculations show that p γ is an pL 1 ; A 1 , A 1 q-local quasi-geodesic in the hyperbolic space p X for some A 1 " A 1 pλ, ǫ, Dq . By increasing L 1 , we can use the local-to-global property of quasi-geodesics in the hyperbolic space p X, to obtain that p γ is a pA 1 , A 1 q-quasi-geodesic where A 1 " A 1 pλ, ǫ, Dq.
Proof. Let f : ra 1 , d 1 s Ñ ra, ds be the homeomorphism of γ 1˚γ2 so that pγ 1˚γ2 q˝f is a pk, cqquasi-geodesic. Let γ " γ 1˚γ2 and s, t P ra, ds. Let s 1 , t 1 , b 1 P ra 1 , d 1 s so that s " f ps 1 q, t " f pt 1 q, and b " f pb 1 q. Without loss of generality, we can assume s P ra, bs and t P rb, ds. The fact that d`γpsq, γptq˘ď k|s´t|`2c follows from the triangle inequality. The other inequality is the following calculation.
We now apply Corollaries 5.8 and 5.9 to finish the proof Corollary 5.12.

5.4.
Deep points decomposition of local quasi-geodesics. As a consequence of the isolation and quasiconvexity of the peripheral subsets (Lemmas 5.3 and 5.5), any quasi-geodesic γ in X has a decomposition γ " σ 0˚α1˚σ1˚¨¨¨˚αn˚σn so that each α i is "deep" in a peripheral, i.e., α i runs close to a single peripheral for a long time, and each σ i avoids getting close to any peripheral for a significant length of time (see [Hru10] and [Sis] for details). The goal of this section is to give a similar decomposition for local quasi-geodesics in X. We begin by defining when an element t of the domain of a local quasi-geodesic γ is "deep" in a peripheral P P P. This means that we can find points s 1 , s 2 on either side of t whose distance from t is between θ and L{4 and with γps 1 q,γps 2 q close to P .
Definition 5.15 (Deep points). Let X be hyperbolic relative to a collection P of peripheral subsets and γ : I Ñ X be an pL; λ, ǫq-local quasi-geodesic where L ě 12θ. A number t P I is P -deep for some P P P, if there are s 1 ă t ă s 2 in I such that θ ď |s i´t | ď L{4 and dpγps i q, P q ď rR. We say s 1 (resp. s 2 ) is a left (resp. right) witness of t. For each P P P, define deeppγ; P q to be the set of P -deep elements of I.
The next three results establish that deeppγ; P q is a collection of disjoint intervals and isolated points of the domain of γ and that points in the domain of γ cannot be deep for two different peripherals.
Proof. Suppose there exists t P deeppγ; P q X deeppγ; U q where P ‰ U and let F : r0, 8q Ñ r0, 8q be the function from Lemma 5.3 that bounds the coarse intersection of P and U . Let rp 1 , p 2 s and ru 1 , u 2 s be the intervals containing t provided by Lemma 5.16 for P and U respectively. Without loss of generality, assume p 1 ď u 1 . Thus, γpu 1 q, γptq P N rR pP q X N rR pU q and dpγptq, γpu 1 qq ě θ 2λ´ǫ ě F prRq by choice of θ. However, this contradicts Lemma 5.3, so we must have deeppγ; P qXdeeppγ; U q " H.
Corollary 5.18. Let γ : I Ñ X be an pL; λ, ǫq-local quasi-geodesic with L ě 12θ. If rt 1 , t 2 s Ď I so that |t 1´t2 | ď L{4 and t 1 , t 2 P deeppγ; P q for some P P P, then we have rt 1 , t 2 s Ď deeppγ; P q. In particular, for each P P P, deeppγ; P q is a (possibly empty) union of disjoint intervals and isolated points of I where the distance between any two connected components of deeppγ; P q is greater than L{4.
Corollaries 5.17 and 5.18 establish that the domain of a local quasi-geodesic can be decomposed into non-overlapping deep points for peripheral subsets and points that are not deep for any peripheral. However, we want to ensure that the deep parts of our decomposition run near the peripheral for a sufficiently long time. To achieve this, we define the following relevant decomposition of a local quasi-geodesic.
Definition 5.19 (Relevant subsegments, relevant decomposition). Let B ě 0 and γ : I Ñ X be an pL; λ, ǫq-local quasi-geodesic with L ě 12θ. A subsegment α Ď γ is B-relevant if the parametrized length of α is at least B and there exists P P P so that the domain of α is the closure of a connected component of deeppγ; P q. If Relevant B pγq " tα i u n i"1 is the set of B-relevant subsegments of γ, then Corollaries 5.17 and 5.18, decompose γ into a concatenation: where each σ i does not contain a B-relevant subsegment of γ. We call this decomposition the B-relevant decomposition of γ. For each α i P Relevant B pγq, there exists a single P i P P so that the interior of the domain of α i is a subset of deeppγ; P i q. We call P i the relevant peripheral for α i and the set tP i u n i"1 the relevant peripherals for γ. The next two lemmas give some basic properties of the relevant decomposition of a local quasigeodesic. Lemma 5.20 says the α i of the decomposition run close to the relevant peripheral P i while the σ i have bounded projection (in terms of B) and hence cannot run close to any peripheral subset. Lemma 5.21 says relevant peripherals for "adjacent" α i must be distinct.
Proof. Let L 1 " L 1 pλ, ǫ, Bq and A 1 " A 1 pλ, ǫ, Bq be the constants from Proposition 5.13 so that if η is an pL 1 , λ, ǫq-local quasi-geodesic in X with the λp4B`ǫq-bounded projections, then π˝η is an pA 1 , A 1 q-quasi-geodesic in p X. Let L 2 " L 1`1 2θA 2 1 and γ be an pL 2 , λ, ǫq-local quasi-geodesic with B-relevant decomposition σ 0˚α1˚σ1˚¨¨¨˚αn˚σn and relevant peripherals tP 1 , . . . , P n u. Suppose P i " P i`1 for some 1 ď i ď n´1. Let I be the domain for γ and ra, bs Ď I be the domain of σ i . If |a´b| ď L 2 {4, then Corollary 5.18 would imply ra, bs Ď deeppγ; P i q contradicting the definition of σ i . If |a´b| ą L 2 {4, then π˝σ i paq and π˝σ i pbq are more than 2`2rR far apart in p X as π˝σ i is a pA 1 , A 1 q-quasi-geodesic and L 2 {p4A 1 q´A 1 ą θ ą 2`2rR. However, this contradicts that γpaq, γpbq P N rR pP i q. Therefore, we must have that P i ‰ P i`1 .
A simple, but central application of Lemma 5.20 is establishing that the σ i and α i are Morse quasi-geodesics. Since the σ i have bounded projections, Corollary 5.12 implies, for sufficient local scale, they are Morse quasi-geodesics, regardless of what the peripheral subsets are. When the peripheral subsets have the Morse local-to-global property, then there is a local scale so that the α i will also be Morse quasi-geodesics by virtue of the fact that they run close to the peripheral P i . Corollary 5.22. For all B ą 2θ, there exists constants L 1 " L 1 pλ, ǫ, Bq, A 2 " A 2 pλ, ǫ, Bq, and Morse gauge M " M pλ, ǫ, Bq so that if σ 0˚α1˚σ1˚¨¨¨˚αn˚σn is the B-relevant decomposition of an pL 2 ; λ, ǫq-quasi-geodesic, then each σ i is an pM ; A 1 , A 1 q-Morse quasi-geodesic.
Corollary 5.23. Suppose every P P P has the Φ-Morse local-to-global property. For every Morse gauge M , there are constants L 3 ě 12θ, k, c and Morse gauge N (all depending only on λ, ǫ, and M ) so that for every B ą 2θ, if σ 0˚α1˚σ1˚¨¨¨˚αn˚σn is the B-relevant decomposition of an pL 3 ; M ; λ, ǫq-local quasi-geodesic γ, then each α i is an pN ; k, cq-Morse quasi-geodesics.
Proof. Recall α i Ď N rR pP i q by Proposition 5.20. Since P i has the Φ-Morse local-to-global property, N rR pP i q, equipped with the induced metric from X, has the Ψ-Morse local-to-global property where Ψ ultimately depends only on Φ, λ, and ǫ. Thus, there exist constants L 3 , λ 0 , ǫ 0 and Morse gauge M 0 so that ΨpM, λ, ǫq " pL 3 , M 0 , λ 0 , ǫ 0 q and if γ is an pL 3 ; M ; λ, ǫq-local Morse quasi-geodesic, then each α i is an pM 0 ; λ 0 , ǫ 0 q-Morse quasi-geodesic in N rR pP i q. The distance formula (Theorem 5.7) and Lemma 5.5 then imply each α i is an pN ; k, cq-Morse quasi-geodesic in X where N , k, and c ultimately depend only on λ, ǫ, and M .
Remark 5.24 (Uniformity of Φ). Corollary 5.23 is the key place in the proof of Theorem 5.1 where the Morse local-to-global property of the peripheral subsets is used. The requirement that the Morse local-to-global property is controlled by the same function Φ for each element of P is precisely to ensure that each α i is a Morse quasi-geodesic whose parameters do not depend on which peripherals are relevant. 5.5. Ordered projections of relevant peripherals. The goal of this section is to prove that if σ 0˚α1˚σ1˚¨¨¨˚αn˚σn is the B-relevant decomposition of a local Morse quasi-geodesic γ, then the distance between the projection of the endpoints of γ to any relevant peripheral P i is bounded below the by a linear function of B. We first show each σ i has uniformly bounded projection (independent of B) onto the relevant peripherals immediately before and after σ i (Proposition 5.25). Next, we establish that the relevant peripherals are "linearly ordered" along the local quasi-geodesic, that is, if 1 ď i ă j ă k ď n, the projection of P i onto P k is uniformly close to the projection of P j onto P k (Proposition 5.30). These two facts mean that the projection of P 1 and P n to each other relevant peripheral P i coarsely equal the endpoints of α i . Since each α i is a quasi-geodesic with parametrized length at least B, the distance between π P i pP 1 q and π P i pP n q will be bounded below by a linear function of B for each other relevant peripheral (Corollary 5.31). Finally, we show that the projection of the endpoints of γ onto any P i must agree with the projection of P 1 and P n onto P n (Corollary 5.32).
We begin with proving the projection of σ i onto P i and P i`1 is bounded independent of B.
We first show that the conclusion of Proposition 5.25 holds for the portion of the σ i that is close to its endpoints.
Lemma 5.26. There exists C " Cpλ, ǫq such that the following holds. Let L ě 12θ, B ą 2θ, and σ 0˚α1˚σ1˚. . . α n˚σn be the B-relevant decomposition of an pL; λ, ǫq-local quasi-geodesic γ. If ra i , b i s is the domain of σ i and s i , t i P ra i , b i s so that |a i´si | ď L{4 and |b i´ti | ď L{4, then diam`π P i pγ| ra i ,s i s q˘ď C and diam`π P i`1 pγ| rt i ,b i s q˘ď C whenever P i or P i`1 exists.
Proof. We shall only prove the case of γ| ra i ,s i s as the other case is analogous. Let C " 2λp2θ2 R`ǫ`1q. There exists rv 1 , v 2 s Ď ra i , s i s so that d`π P i pγpv 1 qq, π P i pγpv 2 q˘ě C{2. By Lemma 5.6 there is ru 1 , u 2 s Ď rv 1 , v 2 s such that d`γpu 1 q, π P i pγpv 1 qq˘ď R and d`γpu 2 q, π P i pγpv 2 qq˘ď R.
Since γ| ra i ,s i s is a pλ, ǫq-quasi-geodesic, the choice of C gives us 2θ`1 ď |u 1´u2 | ď L{4. Thus, there exists τ P pu 1 , u 2 q X deeppγ; P i q. By Corollary 5.18, ra i , τ s Ď deeppγ; P i q, but this is a contradiction to the fact that α i is B-relevant for P i . Thus, we must have diam`π P pγ| ra i ,s i s q˘ď C.
To prove the general case of Proposition 5.25, we employ the following lemma of Sisto.
In light of Lemmas 5.26 and 5.27, Proposition 5.25 will follow if we can show that the portion of σ i far from the endpoints satisfies the hypotheses of Lemma 5.27.
Let ra, bs be the domain of σ i´1 . If |a´b| ď L 4 {4, then the result follows from Lemma 5.26. So, suppose there exists c P pa, bq so that |c´b| " L 4 {4. Since Lemma 5.26 bounded the diameter of π P i`σ i´1 prc, bsq˘, our goal is to bound the diameter of π P i`σ i´1 pra, csq˘. By Lemma 5.20, σ i´1 pbq lies in the rR-neighborhood of P i in X, so π˝σ i´1 pbq has distance at most rR`1 from c P in p X. Since π˝σ i´1 is a parametrized pA 1 , A 1 q-quasi-geodesic, the choice of L 4 yields pπ˝σ i´1 q´1`N C 2 pc P q˘Ď pc, bs.
In particular, π˝σ i´1 | ra,cs is an unparametrized pλ 1 , ǫ 1 q-quasi-geodesic of p X that does not intersect the pC 2`1 q-neighborhood of c P . Lemma 5.27 now yields the result.
Corollary 5.23 established that the relevant subsegments of the decomposition are global Morse quasi-geodesics when the peripherals have the Morse local-to-global property. Since the diameter of the projection of σ i´1 and σ i to the relevant peripheral P i are bounded independent of B, we can use the quasi-geodesic α i to produce a linear lower bound on the distance between π P i pσ i´1 q and π P i pσ i q in terms of B.
Corollary 5.28. Suppose each P P P has the Φ-Morse local-to-global property. Let B ą 2θ and M be a Morse gauge. There exist constants L 5 " L 5 pλ, ǫ, B, M q and A 3 " A 3 pλ, ǫ, M q so that if σ 0˚α1˚¨¨¨˚αn˚σn is the B-relevant decomposition of an pL 5 ; M ; λ, ǫq-local Morse quasi-geodesic and P i is the relevant peripheral for α i , then d`π P i pσ i´1 q, π P i pσ i q˘ě 1 A 3 B´A 3 .
Proof. Let L 5 " maxtL 3 , L 4 u`1 where L 3 " L 3 pλ, ǫ, B, M q and L 4 " L 4 pλ, ǫ, B, M q are the constants from Corollary 5.23 and Proposition 5.13 respectively. Let k " kpλ, ǫ, M q and c " cpλ, ǫ, M q be the constants from Corollary 5.23 so that each α i is a pk, cq-quasi-geodesic. If x i and y i are the left and right endpoints of α i respectively, then x i , y i P N rR pP i q by Lemma 5.20. Let K 0 " K 0 pλ, ǫq be the constant from Proposition 5.25 so that the diameter of π P i pσ i´1 q and π P i pσ i q is at most K 0 . Recall, dpx i , y i q ě B{k´c, since α i has parametrized length at least B (Definition 5.19). Thus, we establish the corollary with the following calculation: 1 k B´c ďdpx i , y i q ďd pπ P i px i q, π P i py i qq`2rR`2 ďd pπ P i pσ i´1 q, π P i pσ i qq`2rR`2`2K 0 .
We now pause to define a local scale Λ larger than any of the local scales required by any of the results in this section up until this point. This will allow all of the proceeding results to apply to any pΛ; M ; λ, ǫq-local Morse quasi-geodesic in X. This scale depends on the relevancy constant B from the decomposition of the local quasi-geodesic, the Morse gauge M , our fixed quasigeodesic constants λ and ǫ, and the function Φ governing the Morse local-to-global property of the peripherals.
We now show the second main tool of this section, the linear ordering of the relevant peripheral subsets along the local Morse quasi-geodesic. Proposition 5.30. Suppose every P P P has the Φ-Morse local-to-global property. For each Morse gauge M , there exist constants K 1 " K 1 pλ, ǫq and B 1 " B 1 pλ, ǫ, M q such that the following holds for for any B ě B 1 . Let Λ " Λ Φ pλ, ǫ, B, M q and σ 0˚α1˚σ1˚¨¨¨˚αn˚σn be the B-relevant decomposition of a pΛ; M ; λ, ǫq-local Morse quasi-geodesic γ. If tP 1 , . . . , P n u are the relevant peripherals for γ, then for all 1 ď i ă j ă k ď n π P k pP i q Ď N K 1 pπ P k pP j qq and π P i pP k q Ď N K 1 pπ P i pP j qq .
Proof. We only show the first containment as the second follows analogously.
Assume B ą 2θ. By Lemma 5.21, we know P i´1 ‰ P i and P i`1 ‰ P i . By Proposition 5.25, there exists K 0 " K 0 pλ, ǫq so that diampπ P i pσ i´1 qq ď K 0 and diampπ P i pσ i qq ď K 0 .
By Lemma 5.20, we have σ i´1 X N rR pP i q ‰ H and σ i X N rR pP i q ‰ H.
Using the fact that π P i is pµ, µq-coarsely Lipschitz and Corollary 5.28 we obtain: d`π P i pP i´1 q, π P i pP i`1 q˘ě d`π P i pσ i´1 q, π P i pσ i`1 q˘´2µprR`1q´2K 0 where A 3 " A 3 pλ, ǫ, M q is as in Corollary 5.28. Let K 1 " µR`4µ, and suppose that B 1 ą 2θ is chosen large enough so that for B ě B 1 , (˚) yields d`π P i pP i´1 q, π P i pP i`1 q˘ě Q`2K 1 .
We now proceed by induction on n, the number of terms in the B-relevant decomposition. Assume n " 3, so γ " σ 0˚α1˚σ1˚α2˚σ2˚α3˚σ3 .
Let 1 ď i ă j ă k ď n. If k ă n or 1 ă i, then the induction hypothesis applies to the local quasi-geodesic σ i´1˚αi˚σi˚. . . α k˚σk and we are finished. Thus, we can assume k " n and i " 1. Now, the induction hypothesis applies to σ 1˚α2˚σ2˚. . . α n˚σn and σ 0˚α1˚σ1˚. . . α n´1˚σn´1 . In particular, π P j pP n q Ď N K 1 pπ P j pP j`1 qq and π P j pP 1 q Ď N K 1 pπ P j pP j´1 qq for 2 ď j ď n´1. By the choice of B 1 and (˚), we have d`π P j pP 1 q, π P j pP n q˘ě Q.
As in the base case, this implies every geodesic connecting a point x P P 1 to π Pn pxq passes through N R pP j q, yielding π Pn pP 1 q Ď N µR`4µ pπ Pn pP j qq " N K 1 pπ Pn pP j qq by Lemma 5.4.
The ordering of the peripherals has two immediate consequences when coupled with Proposition 5.25. Each relevant peripheral is in fact distinct and the distance between the projection of two relevant peripherals onto a peripheral between them is bounded below by a linear function of B.
Corollary 5.31. Suppose each P P P has the Φ-Morse local-to-global property and let M be a Morse gauge. There exist B 2 " B 2 pλ, ǫ, M q and A 4 " A 4 pλ, ǫ, M q so that for all B ě B 2 , if Λ " Λ Φ pλ, ǫ, B, M q and tP 1 , . . . , P n u are the B-relevant peripherals of a pΛ; M ; λ, ǫq-local Morse quasi-geodesic, then ‚ P i " P j if and only if i " j; ‚ d`π P j pP i q, π P j pP k q˘ě 1 A 4 B´A 4 whenever 1 ď i ă j ă k ď n. Proof. By Corollary 5.28, there exists B 2 " B 2 pλ, ǫ, M q so that B 2 is larger than the constant B 1 " B 1 pλ, ǫ, M q from Proposition 5.30 and diampP i q ě µ`2K 1`1 0 where K 1 " K 1 pλ, ǫq is the constant from Proposition 5.30. In Lemma 5.21, we showed P i´1 ‰ P i and P i`1 ‰ P i , so π P i pP i´1 q and π P i pP i`1 q both have diameter at most µ (Lemma 5.4). Proposition 5.30 implies π P i pP j q is contained in the K 1 -neighborhood of π P i pP i´1 q or π P i pP i`1 q. Thus, diam`π P i pP j q˘ď µ`2K 1 . If P i " P j for i ‰ j, then π P i pP j q " P i . But, this would be a contradiction as diampP i q ě µ`2K 1`1 0.
The second item follows by combining Corollary 5.28 with Proposition 5.30 and the fact that σ j´1 X N rR pP j´1 q ‰ H and σ j X N rR pP j`1 q ‰ H.
We now use the results of this section to show that the projection of the endpoints of a local quasi-geodesic to a relevant peripheral P i are coarsely equal to the projection of the first and last relevant peripheral on P i . By Corollary 5.31, this implies the distance between the projection of the endpoints is bounded below by a linear function of the relevancy constant B.
Corollary 5.32. Suppose each P P P has the Φ-Morse local-to-global property and let M be a Morse gauge. There exists constants B 3 " B 3 pλ, ǫ, M q and K 2 " K 2 pλ, ǫq so that for all B ě B 3 , if Λ " Λ Φ pλ, ǫ, B, M q and σ 0˚α1˚¨¨¨˚αn˚σn is the B-relevant decomposition of a pΛ; M ; λ, ǫq-local Morse quasi-geodesic, then π P i pσ 0 q Ď N K 2`π P i pP 1 q˘and π P i pσ n q Ď N K 2`π P i pP n q˘for all i P t1, . . . , nu.
Proof. Let K 1 pλ, ǫq and B 1 pλ, ǫ, M q be the constant from Proposition 5.30. We first show that each σ i has a projection onto each P j bounded independently of B.
Proof. Without loss of generality, assume i ă j. By Proposition 5.25, we can assume j ‰ i`1. Let x P σ i and y " π P i`1 pxq and assume B ě B 1 . We first show that d`π P j pxq, π P j pyq˘ď Q for large enough B.
If d`π P j pxq, π P j pyq˘ě Q, then every geodesic in X from x to y passes through the R-neighborhood of P j (Lemma 5.6). By Lemma 5.4, if η is a geodesic in X from x to π P i`1 pxq " y, then diampπ P i`1 pηq˘ď µ. Thus, if d`π P j pxq, π P j pyq˘ě Q, then d`y, π P i`1 pP j q˘ď 2µ`µR. Now, y " π P i`1 pxq P π P i`1 pσ i q and π P i`1 pσ i q is contained in the pK 0`µ rR`µq-neighborhood of π P i`1 pP i q by Proposition 5.25. Therefore, if d`π P j pxq, π P j pyq˘ě Q, then d`π P i`1 pP i q, π P i`1 pP j q˘ď K 0`5 µrR. However, Corollary 5.31 provides a constant B 2 " B 2 pλ, ǫ, M q, so that for all B ě B 2 , the distance between π P i`1 pP i q and π P i`1 pP j q is bounded below by a linear function (also determined by λ, ǫ, M ) of B. Hence, for sufficiently large B, we would have d`π P i`1 pP i q, π P i`1 pP j q˘ą K 0`5 µrR. This contradiction implies d`π P j pxq, π P j pyq˘ď Q for all B ě B 1 where B 1 " B 1 pλ, ǫ, M q Now for any x, z P σ i , we can use the above plus the triangle inequality and the fact that diam`π P j pP i`1 qq˘ď µ to obtain d`π j pxq, π j pzq˘ď 2Q`µ. Now, let B 3 " B 1`B 1 and K 2 " K 1`µ rR`µ where B 1 and K 1 are as in Claim 5. Since, π P i is pµ, µq-coarsely Lipschitz and σ 0 , σ n intersect the rR-neighborhoods of P 1 and P n respectively (Lemma 5.20), Claim 5 implies π P i pσ 0 q Ď N K 2`π P i pP 1 q˘and π P i pσ n q Ď N K 2`π P i pP n q˘. that γ satisfies the hypothesis of Lemma 2.14. Thus, γ is a global pλ 1 , ǫ 1 q-quasi-geodesic where λ 1 and ǫ 1 depends only on λ, ǫ, and M . Since all subsegments γ are also pL; M ; λ, ǫq-local Morse quasi-geodesics, Lemma 5.33 also implies that γ is N -Morse for some N " N pλ, ǫ, M q.