Surface Houghton groups

For every $n\ge 2$, the {\em surface Houghton group} $\mathcal B_n$ is defined as the asymptotically rigid mapping class group of a surface with exactly $n$ ends, all of them non-planar. The groups $\mathcal B_n$ are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphisms of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $\mathcal B_n$. As countable mapping class groups of infinite type surfaces, the groups $\mathcal B_n$ lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $\mathcal B_n$ is of type $F_{n-1}$, but not of type $FP_n$, analogous to the braided Houghton groups.


Introduction and results
For n ě 2, denote by Σ n the connected orientable surface with exactly n ends, all of them non-planar.We view Σ n as constructed by first gluing a torus with two boundary components (called a piece) to every connected component of a sphere with n boundary components, and then inductively gluing a piece to every boundary component of the surface from the previous step.The surface Houghton group, B n , is the subgroup of the mapping class group MappΣ n q whose elements eventually send pieces to pieces, in a trivial manner; see Section 2 for a precise definition.
Viewed in this light, the groups B n are natural analogs of the asymptotic mapping class groups of Cantor surfaces considered in [16,18,2,19,1].In addition, these groups are closely related to Houghton groups and their braided relatives [9].In fact, using Funar's description of the braided Houghton group brH n as an asymptotically rigid mapping class group [15], in Remark 2.2 below we briefly explain how to see that brH n may be realized as a subgroup of B n .
Beyond this analogy, the groups B n arise naturally in the study of depth-1 foliations of closed 3-manifolds.More precisely, the nonproduct components are mapping tori of end-periodic homeomorphisms (see [14]), and every such homeomorphism is conjugate into some B n (see [13,Corollary 2.9]).
1.1.Finiteness properties.Recall that a group is of type F d if it has a classifying space with finite d-skeleton, and that it is of type FP d if the integers, regarded as a trivial module over the group, have a projective resolution that is of finite type in dimensions up to d.Such a resolution can be obtained for a group of type F d by using the cellular chain complex of the universal cover of a classifying space.Thus, F d implies FP d .
In [19], Genevois-Lonjou-Urech proved that the braided Houghton group brH n is of type F n´1 but not of type FP n .Our first result is the analog of this result in our setting.
Theorem 1.1.B n is of type F n´1 but not of type FP n .
In order to prove Theorem 1.1, and as is often the case with this type of result, we will make use of a classical criterion of Brown [7], expressed through the language of discrete Morse theory; see Section 3 for details.More precisely, for each n ě 2 we will construct a finite-dimensional contractible cube complex on which B n acts, and an invariant discrete Morse function on the complex, such that the descending links are highly connected.In the construction of the complex as well as in the analysis of descending links, we make heavy use of methods developed in [1].1.2.Abelianization.A well-known theorem of Powell [29] asserts that the pure mapping class group PMappSq of a finite-type surface S of genus at least three has trivial abelianization.For the surfaces Σ n , a result of Patel, Vlamis, and the first author [4, Corollary 6] implies that H 1 pPMappΣ n q, Zq -Z n´1 .More generally, the abelianization of (pure) mapping class groups of infinite type surfaces is more complicated; see [10,11,24,26,30].Using [4], we compute the abelianization B ab n of B n , as well as that of their pure counterparts P B n .Theorem 1.2.For all n ě 2, B ab n " t0u and P B ab n -Z n´1 .
From the description of the abelianizations, we also describe all finite quotients of B n , proving that they are highly constrained; see Proposition 6.2.This has the following consequence.
Corollary 1.3.For all m, n P N, the groups B n and brH m are not commensurable.
1.3.Marking graphs.By Theorem 1.1, B n is finitely generated for all n ě 2, and so it has a well-defined coarse geometry.In the finite type setting, a quasi-isometric model for the mapping class group which has proven quite useful is Masur and Minsky's marking graph [25].For the surfaces Σ n , the marking graph is no longer a good model for several reasons.A trivial issue is that it is disconnected, and the orbit of a marking may lie in different components.There are B n -invariant components, but even these are problematic since they are no longer locally finite.Worse, the orbit map to any such component fails to be a quasi-isometric embedding (see Section 7).On the other hand, there are (many) locally finite subgraphs which do serve as quasi-isometric models, by the following and the Milnor-Švarc Lemma (see e.g.[6]).
Theorem 1.4.For all n ě 2, there exists locally finite subgraphs of the marking graph on which B n acts cocompactly.For n ě 3, any marking µ is contained in such a subgraph.
The final statement is not quite true for n " 2 since there are markings on Σ 2 with infinite stabilizer in B n .
Plan of the paper.Section 2 contains the relevant background on surfaces, mapping class groups, and the definition of B n , and in Section 3 we recall a classical result of Brown [7] about finiteness properties of groups, and describe the relevant tools in our setting.Section 4 constructs a contractible cube complex on which B n acts nicely, and which is similar to that of [19,1].In Section 5 we will prove Theorem 1.1.We then prove Theorem 1.2 and Corollary 1.3 in Section 6 and Theorem 1.4 in Section 7.
Acknowledgements.The authors would like to thank Anne Lonjou for helpful conversations.

Surfaces
Throughout this paper, all surfaces will be assumed to be connected, orientable and second-countable.A surface is said to have finite type if its fundamental group is finitely generated; otherwise, it has infinite type.In this paper, the primary surfaces of infinite type we will consider are those that have a finite number of ends, all non-planar; see Section 2.3 for an explicit construction.
2.1.Curves and arcs.By a curve on a surface S we mean the isotopy class of a simple closed curve on S. All curves will be essential, meaning they do not bound a disk or once-punctured disk, nor do they cobound an annulus with a component of BS.An arc is the isotopy class (relative to BS) of an embedded path that connects two boundary components of S. We say that two curves/arcs are disjoint if they can be isotoped off each other.As usual, we will not distinguish between curves (resp.arcs) and their representatives.

Mapping class groups.
The mapping class group MappSq of S is the group of isotopy classes of orientation-preserving homeomorphisms of S; here all homeomorphisms and isotopies are assumed to fix the boundary of S pointwise.The pure mapping class group PMappSq is the subgroup of MappSq whose elements fix every end of S. When S has finite type, then MappSq is well-known to be of type F 8 [20].On the other hand, when S has infinite type, then MappSq is uncountable.
When S has infinite type, an important subgroup of MappSq (and of PMappSq, in fact) is the compactly-supported mapping class group Map c pSq, which consists of those mapping classes that are the identity outside some compact subset of S. By a result of Patel-Vlamis [28], Map c pSq is dense (in the compact-open topology) inside PMappSq if and only if S has at most one non-planar end; otherwise, PMappSq is topologically generated by Map c pSq plus the set of handle-shifts.

2.3.
Rigid structures and surface Houghton groups.The goal of this subsection is to describe the surfaces Σ n and the additional structure eluded to in the introduction necessary to define their asymptotic mapping class groups.These definitions, along with the toolkit needed to present it, closely follows [1, Section 3].
Fix an integer n ě 2. Let O n be a sphere with n boundary components, labelled b 1 , . . ., b n , and T a torus with two boundary components, denoted B ´T and B `T ; we refer to B `T as the top boundary component of T .We fix, once and for all, an orientation-reversing homeomorphism λ : B ´T Ñ B `T , and orientation-reversing homeomorphisms µ i : B ´T Ñ b i , for i " 1, . . ., n.We construct a sequence of compact, connected, orientable surfaces pM i q i as follows: is the result of gluing a copy of T to each boundary component of O n , using the homeomorphisms µ i ; ‚ For each i ě 3, M i is the result of gluing a copy of T along each of the boundary components of M i´1 , using the homeomorphism λ.The surface Σ n is the union of the surfaces M i above.The closure of each of the connected components of M i M i´1 , for i ě 2, is called a piece.By construction, each piece B Ă M i M i´1 is one of the glued on copies of T , and so is equipped with a canonical homeomorphism i B : B Ñ T .We call the unique boundary component of B that belongs to M i´1 the top boundary component of B as it maps by i B to B `T .The set tι B : B is a pieceu is called the rigid structure on Σ n .
A subsurface of Σ n is suited if it is connected and is the union of O n and finitely many pieces.A boundary component of a suited subsurface is called a suited curve.
Let f : Σ n Ñ Σ n be a homeomorphism.We say that f is asymptotically rigid if there exists a suited subsurface Z Ă Σ n , called a defining surface for f , such that: ‚ f pZq is a suited subsurface, and ‚ f is rigid away from Z, that is, for every piece B Ă Σ n Z, we have that f pBq is a piece, and f | B " ι ´1 f pBq ˝ιB .
Definition 2.1 (Surface Houghton group).The surface Houghton group B n is the subgroup of the mapping class group MappΣ n q whose elements have an asymptotically rigid representative.
We will denote by P B n the intersection of B n with the pure mapping class group PMappΣ n q.
Remark 2.2.In [15], Funar described the braided Houghton groups brH n as asymptotically rigid mapping class groups of certain planar surfaces.Replacing punctures of this planar surface with boundary components, and then doubling the resulting surface, determines a homomorphism brH n Ñ B n analogous to the construction Ivanov-McCarthy [23, Section 2].

Finiteness properties
3.1.Brown's criterion for finiteness properties.In this section we recall a classical criterion, due to Brown [7], for a group to (not) have certain finiteness properties.We remark that our formulation of the criterion differs from the original [7], as we use the language of discrete Morse theory as developed in [5].We recall here the basic notions and definitions, referring the interested reader to [5] or [1, Appendix A] for a thorough discussion.
Our setting is a group G acting on a piecewise euclidean CW-complex X by cell-permuting homeomorphisms that restrict to isometries on cells.A Morse function on X is a cell-wise affine map h : X Ñ R satisfying the condition that on each closed cell it attains a unique maximum (at a vertex, the top vertex of the cell).The descending link lk Ó pvq of a vertex v is the part of its link spanned by the links of cells that contain v as their top vertex.The value hpvq is often referred to as the height of v.
We consider the vertices the critical points in X and the images of vertices under h are the critical values.A Morse function h : X Ñ R discrete if the set of its critical values is a discrete subset of R. Now the first theorem can be stated as follows.
Theorem 3.1 (Brown).Let G be a group acting by cell-wise isometries on a contractible piecewise euclidean CW-complex X. Assume X is equipped with a discrete G-invariant Morse function h : X Ñ R, and let X ďs denote the largest subcomplex of X fully contained in the preimage h ´1p´8, ss.Suppose that ‚ The quotient of X ďs by G is finite for all critical values s. ‚ Every cell stabilizer is of type F 8 .‚ There exists d ě 1 such that, for sufficiently large critical values s and for every vertex of v P X with hpvq ě s, the descending link of v in X is d-spherical, i.e., pd ´1q-connected and of dimension d. ‚ For each critical value s, there exists a vertex v with noncontractible descending link at height hpvq ě s.
As the set of critical values is discrete, we can index them in order.Now, we consider the filtration of X by sublevel complexes X i :" X ďs i where s i runs through the critical values in order.The filtration step X i`1 is obtained from X i up to homotopy by coning off descending links.Thus, the hypothesis that descending links are pd ´1q-connected implies that the inclusion of X i into X i`1 induces isomophisms in homotopy (and homology) groups up to dimension d ´1 and an epimorphism in homotopy (and homology) in dimension d.As we assume the complex X to be contractible, the isomophisms in dimensions below d must eventually all be trivial maps.It follows that for sufficiently large i, the sublevel complex X i is pd ´1q-connected.By [7, Propositions 1.1 and 3.1] implies that G is of type F d .
For the negative direction, we focus on the system tH d pX i qu i of homology groups in dimension d.We already observed that for i large enough, the morphisms in the system are onto.Hence, the system can only be essentially trivial if the homology groups H d pX i q vanish for all large enough i. If, however, at the transition from X i to X i`1 , we encounter a non-contractible pd ´1q-connected descending link of dimension d, this descending link will have non-trivial homology in dimension d.Hence the descending link contains a d-cycle.If this cycle was a boundary in X i , coning off the descending link would provide another way of bounding it in X i`1 , thus creating a non-trivial pd `1qcycle, which is impossible as that element of H d`1 could never be killed in the future as we only ever cone off d-dimensional links.Hence, H d pX i q ‰ 0 for infinitely many i.
It is clear from the criterion that tools for the analysis of connectivity properties of spaces can be useful.We collect the tools that we will need in this case.

3.2.
Propagating connectivity properties.We use the convention that every space is p´2q-connected and that any non-empty space is p´1q-connected if it is non-empty.
3.2.1.Complete joins.We will deduce connectivity properties of complexes from those of other complexes, and maps between them.One way to do this is formalized through the notion of a complete join complex, introduced by Hatcher and Wahl in [22].Definition 3.2 (Complete join complex).Let Y and X be simplicial complexes, and π : Y Ñ X a simplicial map.We say that Y is a complete join complex over X (with respect to the map π) if the following properties are satisfied ‚ π is surjective, and is injective on individual simplices.‚ For each simplex σ " xv 0 , . . ., v d y, its preimage can be written as a join of fibers over vertices: Since π is injective on simplices, the π-preimages of vertices are discrete sets of vertices in Y .They are non-empty since π is surjective.It follows that for each d-simplex σ in X, the π-preimage π ´1pσq is a join of d `1 non-empty discrete sets.
We make use of a complete join in two places, once to transfer known connectivity properties from X to Y , and once to go the other way.The first direction is the difficult one, and has been established by Hatcher-Wahl [22].
Before stating the result, recall that a simplicial complex is weakly Cohen-Macaulay of dimension k `1 if it is k-connected and if the link of every simplex σ is pk ´dimpσq ´1q-connected.
Going forward is the easy direction (and is implicit in the argument given by Hatcher and Wahl [22]).See [1,Remark A.15] for an explicit account.
Proposition 3.4.Suppose π : Y Ñ X is a complete join.Then X is a retract of Y and inherits all properties that can be expressed by the vanishing of group-valued functors or cofunctors.In particular, if Y is k-connected, then so is X.
Similarly, complete joins are easily prevented from being contractible.Observation 3.5.Let π : Y Ñ X be a complete join, and suppose that there is a top-dimensional simplex σ " xv 0 , . . ., v d y in X such that each vertex fiber π ´1pv i q contains at least two points.Then, the fiber over σ contains a d-sphere which defines a cycle in the homology of Y that cannot be a boundary as Y does not contain simplices of dimension d `1.In particular, Y is not contractible.

The bad simplex argument.
A map from a complete join is a particularly nice projection.The bad simplex argument, introduced by Hatcher and Vogtmann [21], uses the inclusion map of a subcomplex together with additional local information to transfer connectivity properties from the ambient complex to the subcomplex.We do not follow the original exposition of the argument, since we find the language introduced in [1] more convenient.
Let X be a simplicial complex and assume that we are given a map σ Þ Ñ σ that assigns to each simplex σ in X a (possibly empty) face, σ, of σ.We assume that the following two conditions are satisfied: We call the simplex σ good if σ is empty and bad if σ " σ.Note that by monotonicity, the good simplices in X form a subcomplex X good , which we call the good subcomplex.
The good link of a bad simplex σ is the geometric realization of the poset of those proper cofaces τ ą σ for which τ " σ " σ holds, i.e., lk good pσq :" The following proposition is due to Hatcher-Vogtmann [21, Section 2.1], but this wording is taken from [1, Proposition A.7].
Proposition 3.6.Assume that for some number k ě ´1 and every bad simplex σ, the good link lk good pσq is pk ´dimpσqq-connected.Then the inclusion X good ãÑ X induces isomorphisms in π d for all d ď k and an epimorphism in π k`1 .

A contractible cube complex
The strategy for proving Theorem 1.1 is well-known, and is similar to that used in [8,19,1], for instance.Namely, we will construct a contractible complex X on which B n acts in such way that we can apply Brown's [7] criterion described in Section 3 above.
We now proceed to do this.The definitions of the main objects are the word-for-word adaptations of the ones in [1], and the only differences with our situation will occur when we analyze the connectivity properties of descending links.For this reason, we will briefly recall the constructions from [1, Section 5], referring to that article for a more thorough presentation.
Consider ordered pairs pZ, f q, where Z Ă Σ n is a suited subsurface and f P B n .Two such pairs pZ 1 , f 1 q and pZ 2 , f 2 q are said to be equivalent if f ´1 2 ˝f1 pZ 1 q " Z 2 and f ´1 2 ˝f1 is rigid in the complement of Z 1 .Intuitively, the equivalence class of pZ, f q records the "non-rigid" behavior of f outside Z.For example, if f P B n is the identity outside a suited subsurface Z, then pZ, f q is equivalent to pZ, idq.As another useful example to keep in mind, observe that pZ, f 1 q and pZ, f 2 q are equivalent if f ´1 2 ˝f1 leaves Z invariant and is rigid outside Z.
We denote the equivalence class of pZ, f q by rZ, f s, and the set of all equivalence classes by S. The group B n acts on S, by setting g ¨rZ, f s " rZ, g ˝f s.
We define the complexity of a pair pZ, f q as above to be the genus of Z.Alternatively, observe that since Z is a suited subsurface, it is the union of O n and some number of pieces.As each piece contributes 1 to the genus, the complexity is simply the number of pieces in Z.
Clearly, equivalent pairs have equal complexity, and the action preserves complexity, so we have a B n -invariant complexity function Given vertices x 1 , x 2 P S, we say that x 1 ă x 2 if there are representatives pZ i , f i q of x i , for i " 1, 2, so that f 1 " f 2 , Z 1 Ă Z 2 , and Z 2 Z 1 is a disjoint union of pieces.We stress that ă is not a partial order.
The relation ă can be used to construct a cube complex X for which S is the 0-skeleton.Given x 1 ă x 2 , the set tx | x 1 ĺ x ĺ x 2 u are the vertices of a d-cube, with d " hpx 2 q ´hpx 1 q.We call x 1 the bottom of the cube, as it uniquely minimizes complexity over all its vertices.Since Σ n has exactly n ends, the complex X is n-dimensional.
Observe that the action B n on S preserves the cubical structure, and that the complexity function h extends linearly over cubes to a B n -invariant complexity function (of the same name) For k ě 1, write X ďk for the subcomplex of X spanned by those vertices with complexity ď k.A direct translation of the arguments of Proposition 5.7 and Lemmas 6.2 and 6.3 of [1] yields the following: Theorem 4.1.The cube complex X is contractible, and the action of B n on X satisfies: ‚ Let C be a cube with bottom vertex x " rZ, f s.Then the B nstabilizer of C is isomorphic to a finite extension of MappZq.
In particular, every cube stabilizer is of type F 8 .‚ For every k ě 1, the quotient of X ďk by B n is compact.In light of the theorem above, in order to apply Brown's Theorem 3.1, we need to prove that descending links have the correct connectivity properties.As was the case in [1], the connectivity properties of descending links are determined by those of piece complexes, whose definition we now recall: Definition 4.2 (Piece complex).Let Z be a compact surface with boundary, and let Q be a collection of boundary circles.We define the piece complex PpZ, Qq to be the simplicial subcomplex of the curve complex of Z whose vertices are separating curves which, together with a boundary circle from Q, bound a genus 1 subsurface.If Q " BZ, we will write PpZ, Qq " PpZq.
The relation between the two complexes is encapsulated by the following result, whose proof is exactly the same as that of [1,Proposition 6.6].
Proposition 4.3.Let x " rZ, ids be a vertex of X.Then, the descending link lk Ó pxq is a complete join over the piece complex PpZq.
An immediate consequence of Propositions 4.3 and 3.3 is the following.Before we end this section, we will need a little more information about this complete join for the proof of the negative part of Theorem 1.1.To explain, we first recall that for x " rZ, ids, the complete join map η : lk Ó pxq Ñ PpZq is defined as follows.Given rW, gs P lk Ó pxq, there is a piece Y Ă Σ n so that W Y Y is a suited subsurface and rW Y Y, gs " x " rZ, ids.It follows that gpW Y Y q " Z and gpY q is thus a vertex of PpZq.We then define ηprW, gsq " gpY q.With this, we now state the lemma we will need.Lemma 4.5.For every vertex X P PpZq, the fiber in lk Ó pxq is infinite.
Proof.Given X " ηprW, gsq " gpY q as above, we need to show that η ´1pX q is infinite.For this, we let h : Y Ñ Y be any homeomorphism representing an element of MappY q.We extend h by the identity outside Y to a homeomorphism of the same name, which thus represents an element of B n .Observe that g ˝hpW Y Y q " gpW Y Y q " Z and g ˝hpY q " gpY q Ă Z, while g ˝h is rigid outside Z " W Y Y , so rW, g ˝hs " rZ, ids, and thus rW, g ˝hs P lk Ó pxq with ηprW, g ˝hsq " g ˝hpY q " gpY q " X.That is, rW, g ˝hs P η ´1pX q is another vertex.Moreover, rW, g ˝hs " rW, gs if and only if g ´1 ˝g ˝h " h is rigid outside W (up to isotopy).Since Y is a piece outside W , this can only happen if h is isotopic (in Σ n ) to a homeomorphism which restricts to the identity in Y .This is only possibly if the original homeomorphism h of Y represents the identity in MappY q, modulo Dehn twisting in the essential component of BY in Z (which can be "absorbed" into W ). Since MappY q modulo the (central) subgroup generated by Dehn twisting in this component of BY is infinite, it follows that η ´1pX q is infinite, as required.

Connectivity properties of piece complexes
In this section, we shall establish connectivity properties of piece complexes and finally deduce the finiteness properties of B n .
Let us first observe that this implies that piece complexes are Cohen-Macaulay.
Proof.Observe that the link of a d-simplex σ in PpZ, Qq is isomorphic to the piece complex PpZ 1 , Q 1 q where Z 1 is obtained from Z by cutting off the pieces in σ and Q 1 is the set of those boundary circles in Q that still exist in Z 1 .Then, gpZ 1 q " gpZq´pdimpσq`1q ě 2k`3´dimpσq´1 ě 2pk´dimpσq´1q`3 and |Q 1 | " |Q| ´pdimpσq `1q ě pk ´dimpσq ´1q `2.This implies the link of σ is pk ´dimpσq ´1q-connected, in view of Theorem 5.1, as required.
To analyze the connectivity properties of piece complexes, we shall introduce two more complexes: the injective tethered handle complex T H 1 pZ, Qq, which we will see is a complete join over PpZ, Qq; and the tethered handle complex T HpZ, Qq, which contains the injective tethered handle complex as a subcomplex.As before, if Q " BZ we will simply write T H 1 pZ, Qq " T H 1 pZq and T HpZ, Qq " T HpZq.A diagram of the maps we use reads as follows:

PpZq
We have used the left arrow to pull back connectivity from the piece complex to the descending link.We shall use the middle arrow to push forward connectivity from the injective handle complex to the piece complex, and we will use the inclusion of T H 1 pZq into T HpZq to apply a bad simplex argument, pulling back connectivity from the tethered handle complex to the injective handle complex.The connectivity of the tethered handle complex itself has been analyzed in [1].
5.1.Tethered handle complexes.Let Z denote a compact connected orientable surface with boundary.By a handle in Z we mean a subsurface of Z that avoids the boundary BZ and is homeomorphic to a one-holed torus.Given a handle T , consider (the isotopy class of) a simple arc α that joins BT to a component b Ă Z.We remark that the isotopy class of α is not taken relative to its endpoints, as is sometimes the case.Observe that the regular neighborhood of T Y α Y b is a piece.We will refer to the union of T and α as a tethered handle tethered to b with handle T and tether α.Definition 5.3 (Tethered handle complex).Let Z be a compact orientable connected surface, and let Q be a collection of boundary circles of Z.The tethered handle complex T HpZ, Qq is the simplicial complex whose d-simplices are sets of d `1 pairwise disjoint tethered handles, each tethered to an element of Q.
The following is Lemma 10.12 of [1], which builds upon work of Hatcher-Vogtmann [21]: Lemma 5.4.The tethered handle complex T HpZ, Qq is k-connected, provided that Q is not empty and gpZq ě 2k `3.
We consider the following subcomplex of the tethered handle complex.Definition 5.5 (Injective tethered handle complex).The injective tethered handle complex T H 1 pZ, Qq is the subcomplex of T HpZ, Qq consisting of those simplices where the involved handles are tethered to pairwise distinct boundary components in Q.
The reason why we are interested in this subcomplex is that it is a complete join over the piece complex, which allows us to push forward its connectivity.Indeed, we already oberved that a small regular neighborhood of a tethered handle together with its boundary component is homeomorphic to a piece.In this way we obtain a simplicial map π : T H 1 pZ, Qq Ñ PpZ, Qq .The following is [1, Lemma 8.8], and the proof applies verbatim to this setting: Proposition 5.6.The map π : T H 1 pZ, Qq Ñ PpZ, Qq is a complete join.
In particular, if T H 1 pZ, Qq is k-connected, then so is PpZ, Qq by Proposition 3.4.Thus, we have now reduced the problem to analyzing the connectivity properties of the injective tethered handle complex T H 1 pZ, Qq.
Proof.We induct on k starting at k " ´1, which is trivial.For k ě 0, we use the bad simplex argument for the inclusion of T H 1 pZ, Qq into T HpZ, Qq.
Consider a simplex σ in T HpZ, Qq and define σ to consist of those tethered handles in σ that are tethered to the same boundary component as another handle in σ.Note that T H 1 pZ, Qq consists precisely of those simplices σ for which σ is empty, i.e., the good simplices of T HpZ, Qq.See Note that the good link lk good pσq is isomorphic to T H 1 pZ 1 , Q 1 q, where Z 1 is obtained from Z by cutting off, for each boundary circle used in σ, a neighborhood of the boundary circle and all tethers and handles of σ that it meets.The set Q 1 consists of those boundary circles in Q that still exist in Z 1 .See Figure 5.1.Then, we note the following inequalities which allow us to apply the induction hypothesis.
In a bad simplex each used boundary component tethers to at least two handles, whence we also obtain the following estimate: By the induction hypothesis, lk good pσq is therefore pk´dimpσqq-connected.By Proposition 3.6, the inclusion T H 1 pZ, Qq ãÑ T HpZ, Qq induces isomorphisms in π d for all d ď k.As T HpZ, Qq is k-connected by Lemma 5.4, T H 1 pZ, Qq is k-connected.
Even though we do not have the semi-direct product structure in Theorem 6.1 when restricting to P B n , the projection onto Z n´1 from that theorem still defines a surjective homomorphism fitting into a short exact sequence: where pB n q c is the intersection of B n with PMap c pΣ n q, which is precisely the compactly supported elements of B n .Since pB n q c is a direct limit of mapping class groups of compact surfaces, Powell's result [29] implies that P pB n q ab c " t0u, and therefore P B ab n -Z n´1 .At this point, the fact that B ab n " t0u follows from the above and the natural short exact sequence 1 Ñ P B n Ñ B n Ñ Sympnq Ñ 1, where Sympnq is the symmetric group on n elements, and B n Ñ Sympnq comes from the action on the n ends of Σ n .This is because the action can be used to conjugate generators of P B ab n to their inverses.This proves Theorem 1.2.
We observe that pB n q c is a normal subgroup of B n , since the conjugate of a compactly supported homeomorphism is compactly supported.From the homomorphisms above, the quotient G " B n {pB n q c admits a homomorphism to Sympnq with kernel Z n´1 .It is not hard to explicitly construct a splitting of the associated short exact sequence proving that G -Z n´1 ¸Sympnq.We thus have a short exact sequence 1 Ñ pB n q c Ñ B n Ñ pZ n´1 ¸Sympnqq Ñ 1.
On the other hand, the proof of [28,Theorem 4.6] of Patel and Vlamis (which itself relies on a result of Paris [27]) can be applied verbatim to show that pB n q c has no nontrivial finite quotients, proving the following.Proposition 6.2.Every finite quotient of B n factors through the homomorphism to Z n´1 ¸Sympnq.

An application of this and Theorem 1.1 proves Corollary 1.3:
Proof of Corollary 1.3.First observe that if n ‰ m, then Theorem 1.1 and [19,Theorem 5.24] imply that the finiteness properties of the groups are different, and so they cannot be commensurable.
If n " m, and B n and brH n are commensurable, then after passing to the intersection with the finite index pure subgroups, we find finite index subgroups K ă P B n and J ă P brH n so that K -J.We note here that P brH n is the kernel of the corresponding action on the n (non-isolated) ends of the underlying surface, and is not the subgroup consisting of pure braids.
Applying Proposition 6.2, we see that K ab -Z n´1 , and hence J ab -Z n´1 as well.That is, both abelianizations must simply be the restrictions of the abelianization of P B n , and thus also P brH n , respectively.The kernels of the abelianizations must therefore be finite index subgroups of pB n q c and pbrH n q c , respectively.The former has no finite index subgroups, whereas pbrH n q c admits a homomorphism to Z (being the direct limit of compactly supported braid groups), which therefore has infinitley many nontrivial finite index subgroups.This contradiction proves that B n and brH n are not commensurable.

Marking graphs
A marking µ on a surface is a pants decomposition called the base of the marking, basepµq " Ť i α i , together with a choice of transverse curves β i for each α i ; that is, a curve β i so that ipα i , β j q " 0 if i ‰ j and ipα i , β i q " 1 or 2, depending on whether α i and β i fill a one-holed torus or four-holed sphere, respectively.Masur and Minsky [25] define a graph whose vertices are markings and so that two markings differ by elementary moves, which essentially swaps the roles of α i and β i (together with a certain "clean up" operation to ensure the result is again a marking).Let M n denote the marking graph of Σ n .The image of a marking under a mapping class is again a marking, and the definition of elementary move implies that the mapping class group acts on the marking graph.
For a surface S of finite type, its marking graph MpSq is locally finite and the orbit map is a quasi-isometry, since the action is cocompact.However, the action of B n on an invariant component of M n is not cocompact since one can find arbitrarily many distinct homeomorphism types of markings.Moreover, the orbit map is not even a quasi-isometric embedding.Indeed, if µ is a marking, and t i is the Dehn twist in α i P basepµq, then the distance from µ to t i pµq is uniformly bounded, while the distance from the identity to t i tends to infinity as i Ñ 8 (since these are all distinct elements in a finitely generated group).
To prove Theorem 1.4, we need the following.
Lemma 7.1.For all n ě 2 and any marking µ, the stabilizer of µ in B n is either finite, or contains a finite index subgroup that acts on Σ n by covering transformations.In particular, the stabilizer is finite if n ě 3.
Proof.There is a hyperbolic metric on Σ n so that all α i P basepµq have length 1 and all β i meet α i orthogonally.Then any element of the stabilizer of µ acts on Σ n by isometries.The lemma now follows since the isometry group of Σ n is necessarily discrete, and hence finite for n ě 3, and either finite or virtually cyclic acting by covering transformations for n " 2.
It is straightforward to construct B n -invariant components of M n , for all n ě 2 to which the following theorem applies, and which immediately implies Theorem 1.4.Theorem 7.2.For any µ in a B n -invariant component M 0 n Ă M n with finite stabilizer, there is a locally finite subgraph X Ă M 0 n containing µ so that B n acts properly and cocompactly on G.
Proof.We let G be a finite subgraph which is the union of paths connecting µ to its image under each generator from some fixed finite generating set for B n .Further, we assume that each vertex in G has finite stabilizer as well.This is automatic for n ě 3, and is easy to arrange for n " 2. Now set X " B n ¨G.
The fact that B n acts cocompactly on X is immediate, since the G-translates cover X by construction.The only thing we must prove is that X is locally finite.For this, it suffices to show that K tg P B n | g ¨G X G ‰ ∅u is finite.Suppose there exists an infinite sequence of distinct elements tg n u Ă K. Let x n P G be a vertex so that y n " g n ¨xn P G.There are only finitely many vertices of G, and so after passing to a subsequence (and reindexing), x n " x and y n " y for some x, y P G. Thus, g ´1 1 g n ¨x " x for all n, and hence the stabilizer of x is infinite, a contradiction.
Remark 7.3.The utility in proving that MappSq is quasi-isometric to the marking graph for a finite type surface S is that it allows one to provide a coarse estimate for distances in terms of local information via Masur and Minsky's subsurface projections and their distance formula [25].The Dehn twisting examples above imply that one cannot expect a similar distance formula for B n .However, one may wonder if there is some restricted set of subsurfaces for which one can prove a distance formula.Or perhaps there is still a distance formula for all of M n (which simply does not transfer to B n because it is not quasi-isometric to M n ).

Corollary 4 . 4 .
If PpZq is weakly Cohen-Macaulay of dimension k, then so is the descending link lk Ó pZq.

Figure 5 . 1 .Figure 1 .
Figure1.The figure illustrates an example of Z where g " 5 and Q " BZ has three components.There are three tethered handles, which define a 2-simplex σ, while σ is a 1-simplex spanned by the two tethered handles on the left.The surface Z 1 is the shaded subsurface, which has genus 3 and three boundary components, but Q 1 consists of just the two components on the right.