Metric systolicity and two-dimensional Artin groups

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.


Introduction
Artin groups are among most intensively studied classes of groups in Geometric Group Theory. Conjecturally, they possess nice geometric, topological, algebraic, and algorithmic properties, but most of such features are established only for rather restricted subclasses. Even in the case of twodimensional Artin groups such basic questions as solvability of the Conjugacy Problem or the form of the Dehn function have remained open. One, conjectural, approach to many questions concerning Artin groups is showing that they act geometrically on CAT(0) spaces. Such results were established only for a number of rather limited subclasses of Artin groups, for: rightangled Artin groups (RAAGs) [CD95a]; certain classes of 2-dimensional Artin groups [BC02,BM00]; Artin groups of finite type with three generators [Bra00]; 3-dimensional Artin groups of type FC [Bel05]; spherical Artin groups of type A 4 and B 4 [BM10]; 6-strand braid group [HKS16]. Another method of treating Artin groups is finding other non-positive-curvature-like structures describing them. Such approach was successfully carried out e.g. in [AS83,App84,Pri86,Pei96,Bes99]. In [HO17a] the authors undertake similar path showing that Artin groups of large type are systolic, that is, simplicially non-positively curved. This allowed to prove many new results about such groups. In the current article we exhibit a non-positive-curvature-like structure of all two-dimensional Artin groups and all their finitely presented subgroups, and conclude a number of new algorithmic, and large-scale geometric results for those groups.
As the main tool we introduce a new notion of metrically systolic simplicial complex. Roughly speaking, a simply connected flag simplicial complex with a piecewise Euclidean metric on its 2-skeleton is metrically systolic if all essential loops in links of vertices have (angle) length at least 2π (see Section 2 for details). This definition may be treated as a metric analogue of the definition of systolic complex (see e.g. [Che00,JŚ06,Hag03,HO17a]). Our main tool for exploring features of metrically systolic complexes is the use of disc diagrams. It allows us to prove the following results about metrically systolic complexes and groups acting on them geometrically, that is, metrically systolic groups.
Theorem 1.1. Let X be a metrically systolic complex, and let G be a metrically systolic group. Then the following properties hold.
(1) Every loop in X bounds a CAT(0) disc diagram (see Theorems 2. 6 and 2.8 in the text).
(2) The Dehn functions of X and G are quadratic (see Corollary 2.7).
(3) Finitely presented subgroups of G are metrically systolic (see Theorem 3.1). (4) If G is torsion-free and g m is conjugated to g n only when g n = g m , for every g ∈ G, then the Conjugacy Problem is solvable in G (see Theorem 3.6).
We believe that metrically systolic complexes deserve further extensive studies on their own; see a list of open questions in Section 7. Geometrically, metric systolicity enables us to formalize a weaker notion of non-positively curved space where one only requires every minimal filling disc of a 1-cycle to be non-positively curved. This naturally arises by examining the geometry of 2-dimensional Artin groups. It is interesting to compare this with the work of Petrunin and Stadler [PS17], where (roughly speaking) they showed any minimal disc in a CAT (0) space is CAT (0). Thus it is natural to wonder whether one can set up this weaker notion in a more analytical way and apply it to natural classes of examples.
In the current paper we focus on the use of metric systolicity in the context of Artin groups. To this end, starting with the standard Cayley complex for a 2-dimensional Artin group G, we modify it to obtain a metrically systolic G-complex. Therefore, we conclude the following.
We refer to the next subsection for an intuitive explanation of the construction of the complex, as well as comparison with our previous work on constructing systolic complexes for large-type Artin groups from [HO17a].
Direct consequences of Theorem 1.1 and Theorem 1.2 are new results on 2-dimensional Artin groups and their subgroups listed in Corollary 1.3. Let us note that even if 2-dimensional Artin groups were CAT(0), this, a priori, would not say anything about their finitely presented subgroupsthis suggests an important advantage of metric systolicity. Moreover, by Brady and Crisp [BC02], there are 2-dimensional Artin groups which can not act nicely on 2-dimensional CAT(0) complexes. On the other hand, metric systolicity enables us to stay in the 2-dimensional world -one need to study only CAT(0) disc diagrams. This will be convenient for our further work in [HO17b] concerning quasi-isometries of 2-dimensional Artin groups. Corollary 1.3. Let G be a finitely presented subgroup of a 2-dimensional Artin group. Then: (1) G has quadratic Dehn function and, in particular, solvable Word Problem; (2) G has solvable Conjugacy Problem; (3) G has constant filling radius for 2-spherical cycles; (4) Morse Lemma for two-dimensional quasi-discs in G holds.
Dehn function, Word Problem, and Conjugacy Problem are among the most basic notions explored in the context of any group. Still, little was known about them for 2-dimensional Artin groups and their finitely presented subgroups prior to our work.
As far as we know there have been no general results concerning Dehn function for 2-dimensional Artin groups before. Chermak [Che98] proved the Word Problem is solvable for 2-dimensional Artin groups, but no general statement of this type have been known for finitely presented subgroups.
Assertions (3) and (4) from Corollary 1.3 could be derived without referring to metric systolicity. However, for the proof of the strong form of (3), as presented in Theorem 3.7 in the text, the use of metric systolicity is very convenient. This result, in turn, is a crucial ingredient in the proof of the Morse Lemma for two-dimensional quasi-discs (see the proof of Theorem 3.9). The latter is an important large-scale feature of metrically systolic complexes, groups, and of 2-dimensional Artin groups.
The metrically systolic complexes constructed in Theorem 1.2, as well as the large-scale features mentioned above, will play fundamental role in the study of quasi-isometric invariants of 2-dimensional Artin groups in our subsequent work [HO17b]. For applications in [HO17b] we need another result, presented in the following theorem. It does not rely on metric systolicity, and follows from known facts, but it seems that it is not present in the literature.
Comments on the proof of Theorem 1.2. Here we present a rough idea of the construction of metrically systolic complexes for two dimensional Artin groups. Let Γ be a finite simple graph with each of its edges labeled by an integer ≥ 2. An Artin group with defining graph Γ, denoted A Γ , is given by the following presentation. The generators of A Γ are in one to one correspondence with vertices of Γ, and there is a relation of the form aba · · · n = bab · · · n whenever two vertices a and b are connected by an edge labeled by n.

METRIC SYSTOLICITY AND TWO-DIMENSIONAL ARTIN GROUPS 5
An Artin group is of dimension d if it has cohomological dimension d. By Charney and Davis [CD95b], A Γ has dimension ≤ 2 if and only if for any triangle ∆ ⊂ Γ with its sides labeled by p, q, r, we have 1 p + 1 q + 1 r ≤ 1. In particular, the class of all large-type Artin groups, where the label of each edge in Γ is ≥ 3, is properly contained in the class of Artin groups of dimension ≤ 2.
Let A Γ be an Artin group of dimension ≤ 2 and let X * Γ be the presentation complex of A Γ . A natural way to metrize X * Γ is to declare each 2-cell in X * Γ is a regular polygon in the Euclidean plane. However, if we take 2-cells Π 1 and Π 2 (say, two n-gons) such that P = Π 1 ∩ Π 2 is a path with ≥ 2 edges, then any interior vertex of P is not non-positively curved. Let o i be the center of Π i and let the two endpoints of P be v 1 and v 2 . Let K be the region in Π 1 ∪ Π 2 bounded by the 4-gon whose vertices are o 1 , o 2 , v 1 and v 2 . Those positively curved corner points are contained in K. Now we add a new edge e between o 1 and o 2 and add two new triangles {∆ i } 2 i=1 such that the three sides of ∆ i are e, o 1 v i and o 2 v i ; see Figure 1. Figure 1. Adding the edge e and the triangles ∆ 1 = o 1 o 2 v 1 , and ∆ 2 = o 1 o 2 v 2 (in the universal cover of X * Γ ).
Geometrically, one can think of K as a configuration sitting inside the Euclidean 3-space E 3 . Then positively curved points in K give rise to corners in the configuration. Now we use the polyhedron bounded by K ∪∆ 1 ∪∆ 2 to fill these corners. Combinatorially, one can think of ∆ 1 ∪∆ 2 as a replacement of K to get rid of positively curved points in the disc diagram. Now we decide the length of e. From the geometric viewpoint, e should be shorter if P is longer. From the combinatorial viewpoint, we would like o i to be flat after we replace K by ∆ 1 ∪∆ 2 .
4n 2π (|P | is the number of edges in P ), which determines the length of e.
Pick a triangle ∆ ⊂ Γ, then ∆ gives rise to three 2-cells arranged in a cyclic fashion around a vertex v. The condition on two dimensionality of A Γ implies v is already non-positively curved in such configuration, so we do not apply any modifications here.
The main difference between the construction in [HO17a] and the one in this paper is that the former is purely combinatorial, while the current one uses both the metric and combinatorial structure. Thus the method in this paper has more flexibility and applies to a much larger class of Artin groups. Moreover, the structure of flat points in the disc diagrams is more convenient for our later use in [HO17b]. However, since we are now outside the purely combinatorial setting, some results from [HO17a] -e.g. biautomaticity -are much harder to obtain.
Organization of the paper. In Section 2 we define metrically systolic complexes and prove their fundamental property -every cycle can be filled by a CAT (0) disc diagram. In Section 3, we prove the rest of properties in Theorem 1.1, using CAT (0) disc diagrams as a basic tool. In Section 4, we construct the metrically systolic complexes for dihedral Artin groups. In Section 5, we study the local properties of these complexes with two purposes. First we show the complexes in Section 4 are indeed metrically systolic. Second we show that there are no local obstructions to metric systolicity if we glue these complexes together under certain conditions. In Section 6 we glue the complexes for dihedral Artin groups to construct the metrically systolic complexes for any 2-dimensional Artin groups, and prove Theorem 1.2. In Section 7, we prove Theorem 1.4 and leave some open questions about metrically systolic groups and complexes. odowe Centrum Nauki, grants no. UMO-2015/18/M/ST1/00050 and UMO-2017/25/B/ST1/01335. A major part of the work on the paper was carried out while D.O. was visiting McGill University. We would like to thank the Department of Mathematics and Statistics of McGill University for its hospitality during that stay.

Metrically systolic complexes
In this section we introduce the notion of metrically systolic complex. Then we show its most important feature, used later extensively for proving other properties of metrically systolic complexes and groups. The feature is the existence of CAT(0) disc diagrams filling any cycle inside the complex; see Theorem 2.6 and Theorem 2.8. The proofs presented in Subsection 2.2 go by modifying any given disc diagram to a CAT(0) one by performing a finite sequence of local "moves". As an immediate consequence we obtain the quadratic Dehn function in Corollary 2.7.
2.1. Definition. Let X be a flag simplicial complex with its two-skeleton X (2) equipped with a metric d in which every 2-simplex (triangle) is isometric to a Euclidean triangle. For a vertex v ∈ X its link, denoted lk(v, X (1) ), is the full subcomplex (subgraph) of X (1) spanned by all vertices adjacent to v. Every link is equipped with an angular metric, defined as follows.
For an edge v 1 v 2 , we define the angular length of this edge to be the angle ∠ v (v 1 , v 2 ) with the apex v. This turns the link into a metric graph, and the angular metric, which we denote by d ∠ , is the path metric of this metric graph (note that a priori we do not know whether for adjacent vertices v 1 and v 2 ). The angular length of a path ω in the link, which we denote by length ∠ (ω), is the summation of angular lengths of edges in this path. In this paper we assume that the following weak form of triangle inequality holds for angular length in X: for each v ∈ X and every three pairwise adjacent vertices v 1 , v 2 , v 3 in the link of v we have that Then we call X (with metric d) a metric simplicial complex.
Remark 2.1. Note that we allow that the inequality becomes equality -intuitively it corresponds to degenerate 2-simplices in a link, which corresponds to degenerate 3-simplices in X.
For k = 4, 5, 6, . . ., a simple k-cycle C in a simplicial complex is 2-full if there is no edge connecting any two vertices in C having a common neighbor in C.
Definition 2.2 (Metrically systolic complexes and groups). A link in a metric simplicial complex is 2π-large if every 2-full simple cycle in the link has angular length at least 2π. A metric simplicial complex X is locally 2π-large if every its link is 2π-large. A simply connected locally 2π-large metric complex is called a metrically systolic complex. Metrically systolic groups are groups acting geometrically by isometries on metrically systolic complexes.
Remark 2.3. A systolic complex, that is, a connected simply connected flag simplicial complex for which all full cycles in links consist of at least six edges is metrically systolic when equipped with the metric in which all triangles are Euclidean triangles with edges of unit lengths. For more on systolic complexes see e.g. [ A singular disc D is a simplicial complex isomorphic to a finite connected and simply connected subcomplex of a triangulation of the plane. There is the (obvious) boundary cycle for D, that is, a map from a triangulation of 1sphere (circle) to the boundary of D, which is injective on edges. For a cycle C in a simplicial complex X, a singular disc diagram for C is a simplicial map f : D → X from a singular disc D, which maps the boundary cycle of D onto C; see Figure 2 (left). By the relative simplicial approximation theorem [Zee64], for every cycle in a simply connected simplicial complex there exists a singular disc diagram (cf. also van Kampen's lemma e.g. in [LS01,). Below we describe how to obtain singular disc diagrams with some additional properties, by modifying a given one.
A singular disc diagram is called nondegenerate if it is injective on all simplices. It is reduced if distinct adjacent triangles (i.e., triangles sharing an edge) are mapped into distinct triangles. The area of a singular disc diagram is the number of 2-simplices (triangles) in the underlying singular disc. A singular disc diagram for a cycle C in X is minimal if it has the minimal area among singular disc diagrams for C in X. For a metric simplicial complex X and a nondegenerate singular disc diagram f : D → X we equip D with a metric in which f | σ is an isometry onto its image, for every simplex σ in D. Then, f is a CAT(0) singular disc diagram if D is CAT(0), that is, if the angular length of every link in D being a cycle (that is, the link of an interior vertex in D) is at least 2π.
Parallelly to singular disc diagrams one may consider a related notion of singular strip diagrams. A singular strip S is a simplicial complex isomorphic to an infinite connected and simply connected subcomplex of a triangulation of the plane whose complement has two infinite components. The two infinite paths being boundaries of those components are called the boundary paths of S. Having two infinite paths P, P in X, a singular strip diagram for the pair P, P is a simplicial map f : S → X from a singular strip S into X mapping boundary paths of S onto, respectively, P and P ; see Figure 2 (right). A nondegenerate, reduced or CAT(0) singular strip diagram is defined analogously as the corresponding singular disc diagram.
Having a singular disc diagram f : D → X for a cycle C in X we describe a way of producing a new singular disc diagram f : D → X for C, with some additional properties (see e.g. Theorem 2.6 below). In order to do this we need elementary operations -moves -described below.
A-move: Assume there exist pairwise adjacent vertices u, v, w not bounding a triangle in D, that is, there are vertices v 1 , . . . , v k in the region in D bounded by edges between u, v, w. The new singular disc D is obtained from D by removing all the vertices v i (and hence also edges containing them); see Figure 3 (at the top). The new map f : D → X is defined as f (x) = f (x), for all vertices x in D , and then extended simplicially. Such modification is called the A-move on u, v, w and is denoted by A(u, v, w).
For the next moves we assume that the situation as above does not happen, that is, each triple of pairwise adjacent vertices defines a triangle in D.
In particular it means that for each internal edge uv in D there are exactly two vertices w, z each adjacent to both u and v.
B-move: Assume there are two triangles uvw and uvz such that f (w) = f (z). The new singular disc D is obtained from D by removing the edge uv and adding an edge wz; see Figure 3. By our assumptions D is a simplicial singular disc. The new map f : D → X is defined as f (x) := f (x), for all vertices x in D , and then extended simplicially. Such modification is called B-move on u, v and is denoted by B(u, v).

C-move:
Assume there is an edge u 1 u 2 such that f (u 1 ) = f (u 2 ). Such edge need to be internal, so that there are two triangles u 1 u 2 w and u 1 u 2 z containing the edge. The new singular disc D is obtained from D by removing u 1 , u 2 (and all edges containing them), and then adding a new vertex u adjacent to all vertices (of D except u 1 , u 2 ) that are adjacent in D to u 1 or u 2 ; see Figure 3. By our assumptions D is a simplicial singular disc. The new map f : D → X is defined as f (x) := f (x), for all vertices x = u in D , and f (u) := f (u 1 ) = f (u 2 ), and then extended simplicially. Such modification is called C-move on u 1 , u 2 and is denoted by C(u 1 , u 2 ).  The following lemma is essentially the same as [Che00, Lemma 5.1] and [JŚ06, Lemma 1.6]. Although in the latter two only simple cycles are considered, the general case follows by decomposing a given cycle into simple pieces. We omit the straightforward proof.
Lemma 2.4. Let f : D → X be a singular disc diagram for a cycle C in a simplicial complex X. Then by applying A-moves, B-moves, and Cmoves the diagram may be modified to a nondegenerate reduced singular disc diagram for C. In particular, any minimal singular disc diagram for C is nondegenerate and reduced.
The main technical tool for dealing with metrically systolic complexes are CAT(0) singular disc diagrams. Their existence is established in the following theorem. It is an analogue of a result for systolic complexes obtained in [Che00, pp. 159-161] and [JŚ06, Lemma 1.7]. The proof is also an analogue of the systolic case proof. Before the theorem we prove a useful lemma.
Lemma 2.5. Let f : D → X be a singular disc diagram into a metrically systolic complex X. Suppose that there is an interior vertex v in D whose link is a cycle C of angular length less than 2π. Then, by performing a finite number of A-, and D-moves we may find a singular disc diagram f : D → X such that D is a union of the full subcomplex of D spanned by all vertices of D except v, and triangles with vertices in C, and the map f agrees with f on all vertices of D and on all edges coming from D.
Proof. We proceed by induction on the combinatorial length of C. If this length is 3 then we perform A-move. Assume that C consists of at least 4 edges. Denote C = (v 1 , v 2 , . . . , v k ). Then f (C) = (f (v 1 ), f (v 2 ), . . . , f (v k )) is a cycle in X of angular length less then 2π. There is 2 < l ≤ k such that C = (f (v 1 ), f (v 2 ), . . . , f (v l )) is a simple cycle. This is a cycle in the link of f (v) of angular length less than 2π. If l = 3 then f (v 1 ) and f (v 3 ) are adjacent. If l > 3 then, by metric systolicity, C is not 2-full. This means that there exists a vertex, say f (v 2 ), such that its neighbors in C -in our case f (v 1 ) and f (v 3 ) -are adjacent. Hence we may perform D-move D(v, v 2 ), to obtain a new singular disc diagram f : so that the angular length of the link of v in D , being the cycle (v 1 , v 3 , . . . , v k ), is less than 2π. By the inductive assumption we obtain the desired diagram f : D → X.
Theorem 2.6 (CAT(0) disc diagram). Let f : D → X be a singular disc diagram for a cycle C in a metrically systolic complex X. By performing a finite number of A-, B-, C-, D-moves the diagram may be modified to a CAT(0) nondegenerate reduced singular disc diagram f : D → X for C. Furthermore: (1) f does not use any new vertices in the sense that there is an injective map i from the vertex set of D to the vertex set of D such that f = f • i on the vertex set of D; (2) the number of 2-simplices in D is at most the number of 2-simplices in D; (3) any minimal singular disc diagram for C is CAT(0) nondegenerate and reduced.
Proof. We proceed with successive diagrams f : D → X, starting from f := f depending on the following cases.
Case 1: A-move, B-move, or C-move may be performed. Then the new diagram f : D → X is obtained by performing the corresponding move.
Case 2: No A-move, B-move, or C-move may be performed and there exists an internal vertex v whose link is a cycle C = (v 1 , v 2 , . . . , v k ) of angular length less than 2π (in the metric induced from X). Then, by Lemma 2.5 there exists a singular disc diagram f : D → X, where D is obtained from D by replacing the star of v with a disc without internal vertices, and f coincides with f on all vertices except v.
Case 3: We are not in situations from Case 1 or Case 2. Then the diagram f : D → X is a CAT(0) nondegenerate reduced singular disc diagram for C.
After performing modifications as in Case 2, the area of the diagram decreases. Proceeding as in Case 1, that is performing A-moves, B-moves, or C-moves eventually decreases the area of the diagram. It is so because A-move and C-move decrease the area, and after performing B-move we are in position to perform A-move or C-move. Hence eventually we end up in Case 3.
Corollary 2.7. The Dehn function of a metrically systolic complex or group is at most quadratic.
For further applications (e.g. in [HO17b]) we will need singular disc diagrams with some further features (see Theorem 2.8 below). To construct them we have to consider other types of moves: E-moves and F-moves described below. Again, starting from a singular disc diagram f : D → X into a metrically systolic complex X we construct a new diagram f : D → X. For the new moves we assume that we are in the situation when no A-, B-, or C-move may be performed, and there is an interior vertex v and two vertices w, z in its link such that f (w) = f (z). Observe that then w and z are not adjacent.
E-move: Assume that there does not exist a vertex different than v and adjacent to both w, z. We assume furthermore that the angular lengths of two paths between w and z in the link of v are strictly smaller than 2π. The new disc diagram f : D → X is obtained as follows. First we construct an intermediate singular disc D by "collapsing" vertices w, z to a single vertex x, that is, we remove w, z, and introduce a new vertex x adjacent to all vertices that were adjacent in D to w or z; see Figure 4 (top). Furthermore, we add two "copies" v , v of the vertex v, adjacent to vertices in two paths of the link of v, and to x.
, and f agrees with f otherwise. Observe that the angular lengths of links of v and v are strictly smaller than 2π. Hence, by double application of Lemma 2.5 we find a desired singular disc diagram f : D → X with the two links filled without internal vertices.
F-move: Assume that there exists a vertex u different than v and adjacent to both w, z. We first construct a singular disc diagram f : D → X by joining w and z by an edge, removing edges from v "crossing" the new edge wz and adding a copy v of v adjacent to vertices in the original link of v not adjacent to v anymore; see Figure 4 (bottom). In D there is a triangle wzu, and performing the A-move A(u, w, z) we obtain the desired singular disc diagram f : D → X.
Theorem 2.8 (CAT(0) disc diagram II). Let f : D → X be a singular disc diagram for a cycle C in a metrically systolic complex X. By performing a finite number of A-, B-, C-, D-, E-, F-moves the diagram may be modified to a CAT(0) nondegenerate reduced singular disc diagram f : D → X for C satisfying the following property. For every flat vertex v ∈ D the restriction f | St(v) is injective. Furthermore: (1) f does not use any new vertices in the sense that there is an injective map i from the vertex set of D to the vertex set of D such that f = f • i on the vertex set of D; (2) the number of 2-simplices in D is at most the number of 2-simplices in D; (3) any minimal singular disc diagram for C is such.
Proof. By Theorem 2.6, using finitely many A-, B-, C-, D-moves we may modify f to a CAT(0) nondegenerate reduced singular disc diagram f . Moreover, we may reach the situation when no A-, B-, C-move is possible. If for every flat vertex v the restriction f | St(v) is injective then we are done with f = f . If not, we are in a position to perform an E-move or an F-move. Both decrease the area.
Applying iteratively the above procedure we finally obtain the desired singular disc diagram f : D → X. Assertions (1), (2), and (3) follow directly from the construction.
Remark 2.9. Observe that the assertion of the lemma is not true if the vertex is not flat -the star of such vertex could be mapped onto the simplicial cone over a wedge of two cycles. Figure 4. E-move and F-move.
Remark 2.10. We could reduce the number of moves for proving Theorem 2.6 or Theorem 2.8 by allowing singular discs to be non-simplicial, as e.g. in [JŚ06, proof of Lemma 1.6]. We decided to stay in the realm of simplicial complexes.

Properties of metrically systolic complexes and groups
In this section we prove several properties of metrically systolic complexes and groups. In particular, such properties hold for two-dimensional Artin groups, and -as explained in Subsection 3.1 below -for all their finitely presented subgroups.
3.1. Finitely presented subgroups. In this subsection we show that being metrically systolic for groups is inherited by taking finitely presented subgroups. It follows that all subsequent features (and the quadratic isoperimetric inequality established above) of metrically systolic groups are valid also for all their finitely presented subgroups. In particular, they hold for all finitely presented subgroups of two-dimensional Artin groups.
Theorem 3.1. Finitely presented subgroups of metrically systolic groups are metrically systolic.
Proof. In view of [HMP14, Theorem 1.1] (compare also [Wis03, Corollary 5.8]) it is enough to show that the class of locally 2π-large complexes is closed under taking covers and full subcomplexes.
Let X → X be a cover of a locally 2π-large complex X. Then links in X are combinatorially isomorphic to links in X. It follows that such links equipped with a metric induced by the isomorphism are 2π-large. Such metric on links is the angular metric coming from the metric on X induced by the covering. Therefore, X is metrically systolic.
LetX be a full subcomplex of a metrically systolic complex X, equipped with a subcomplex metric. Let C be a 2-full simple cycle in the link of a vertex ofX. By fullness ofX, C is 2-full in X, hence its angular length is at least 2π. Therefore, the angular length of C inX is at least 2π as well. It follows thatX is locally 2π-large.
3.2. Solvability of the Conjugacy Problem. In this subsection we show that the Conjugacy Problem is solvable for torsion-free metrically systolic groups satisfying some additional technical assumption; see Theorem 3.6. The proof is a typical argument for showing solvability of the Conjugacy Problem in the non-positive curvature setting; see e.g. [BH99, Below, and in further parts of the article we use the following convention concerning quasi-isometries.
between metric spaces such that For the rest of the subsection let G be a torsion-free group acting geometrically on a metrically systolic complex X. We will use here the induced metric d in the one-skeleton of X. By scaling the metric we may assume that all edges have length at most 1. Let S be a finite (symmetrized) generating set for G, and let Γ := Cay(G, S) be the corresponding Cayley graph. Let d S be the word metric on G and (the 0-skeleton of) Γ, and |g| S = d S (1 G , g).
The following two lemmas are standard but we formulate them for the purpose of refereeing to constants appearing later. The first one is just the Milnor-Schwarz lemma.
Let D be a planar CAT(0) 2-complex constructed from triangles isometric to triangles in X. Let δ be a CAT(0) geodesic between two given vertices v, u in D. A path δ in the 1-skeleton of D is approximating the geodesic δ if δ is contained in the union of all edges and triangles intersecting δ, and δ is the shortest path with this property. The following is a consequence of e.g. [BH99,Proposition I.7.31].
Lemma 3.4. There exist constants K 2 , L 2 > 1 depending only on the geometry of X (in fact, on the set of isometry types of triangles in X) such that K −1 2 |δ | − L 2 ≤ |δ| ≤ K 2 |δ | + L 2 . Let K := max{K 1 , K 2 } and L = max{L 1 , L 2 }. In particular, it means that the assertions of Lemmas 3.3 and 3.4 hold when the corresponding constants K i and L i are replaced by K and L.
Lemma 3.5. Let g, h ∈ G be conjugate elements, such that, for every vertex v ∈ X, the shortest path between v and gv consists of at least 4 edges. Then there exists an element p ∈ G, conjugating them, that is, g = php −1 , and such that |p| S ≤ A, where A is a constant depending only on |g| S and |h| S (and on the action of G on X).
Proof. For every generator s ∈ S, choose a geodesic 1-skeleton path q(s) in X, between x and sx. Let p be an element conjugating g and h. We will show that starting with p we may find a conjugator p with |p | S ≤ A, where A is a constant depending only on |g| S and |h| S .
Let α S , γ S , and η S be geodesics in Γ between 1 G and, respectively p, g, and h. Let s p 1 · · · s p a , s g 1 · · · s g b , and s h 1 · · · s h c be words in S defined by these geodesics. Let α be the concatenation of paths q(s p 1 ), . Similarly, let γ be the concatenation of paths q(s g 1 ), s g 1 q(s g 2 ), , and let η be the concatenation of paths q(s h 1 ), Consider the cycle C based at x, being the concatenation of (in this order) γ, gα, pη, and α; see Figure 5.
By Lemma 3.3, there exist constants E 1 and F 1 depending only on |g| S and |h| S (and the action of G on X) such that |γ| ≤ E 1 , |η| ≤ F 1 , where | · | denotes the d-length. In what follows we will consider constants depending on E 1 , F 1 , and K, L leading, eventually, to a constant A as in the statement of the lemma.
Let f : D → X be a singular disc diagram for the cycle C. We create a singular strip diagram f : D → X as follows. For every n ∈ Z let D n be a copy of D, and let f n be the simplicial map such that f n (v) := g n f (v), for every vertex v ∈ D -here we identify D n with D. In particular f 0 = f . Next, for every n, we identify the copy of the path gα in D n with the copy of the path α in D n+1 . This way we obtain a singular strip D = n∈Z D n .
We define the map f as the union of maps f n , for all n. This way we obtain the singular strip diagram f : D → X for the pair of paths γ, pη, where γ is the concatenation of paths g n γ, and pη is the concatenation of paths g n pη, for all n ∈ Z; see Figure 5. Observe that there is a g -action on Figure 5. Scheme for proving Lemma 3.5.
D: g n D m = D n+m , and that the map f is equivariant with respect to this action and the g -action on X.
For every m = n, and for each triple of pairwise adjacent vertices v 1 , v 2 , v 3 in D, the A-moves A(g m v 1 , g m v 2 , g m v 3 ) and A(g n v 1 , g n v 2 , g n v 3 ) may be performed independently, since the shortest path between g m v i and g n v j has at least 3 edges. Similarly, B-moves, C-moves, and D-moves may be performed independently for distinct translates of the defining vertices. Thus, we may define an equivariant A-move on u, v, w as the modification consisting of Amoves A(g n u, g n v, g n w), for all n. Similarly we define equivariant B-move, equivariant C-move, and equivariant D-move. As an equivariant analogue of Theorem 2.6 we claim that by performing a finite number of equivariant moves the singular strip diagram f : D → X may be modified to a CAT(0) nondegenerate reduced singular strip diagram f : D → X for the pair γ, pη.
Let β be the CAT(0) geodesic in D with endpoints x, px (that is, their preimages in D ). Let d D denote the CAT(0) distance in D . Since d D (x, gx) ≤ |γ| ≤ E 1 , and d D (x, hx) ≤ |η| ≤ F 1 , by the CAT(0) geometry, and the g -invariance of D , we have . . , v r = px) be a path in the 1-skeleton of D with endpoints x, px (that is, their preimages in D ) approximating the CAT(0) geodesic in D between x and px. Then gβ approximates gβ , and hence, for every vertex v of β, we have where y ∈ β is a point closest to v. Using Lemma 3.4 we get 3). Additionally, we set g 0 = 1 G and g r = p. Then, by Lemma 3.3, for every i = 0, 1, . . . , r − 1 we have (3) By (1) and (2), we have For every i we choose a d S -geodesic between g i and g i+1 . Let β S be their concatenation. This is a path in Γ connecting 1 G and p. By (3) and (4), for every a ∈ β S , we have where g i is the closest to a among g i 's. Now consider the quadrilateral Q in Γ formed by paths γ S , β S , pη S , gβ S . For every vertex v on β S pick a geodesic γ v between v ∈ β S and gv ∈ gβ S . There are at most A := |S| L+1 different up to G-action on Γ paths of length less than L. Hence if |β S | S > A then there are two vertices v, v ∈ β S such that the two paths γ v and γ v are the same up to G. Cutting Q along such paths and gluing together we obtain a quadrilateral Q formed by paths γ S , β S , pη S , gβ S , and such that again d S (a, ga) ≤ L, for all a ∈ β S . This way we construct a quadrilateral Q consisting of paths γ S , β S , pη S , gβ S , with |β S | S ≤ A. Hence we obtain an element p ∈ G conjugating g and h, with |p | S ≤ A.
Theorem 3.6. Let G be a torsion-free metrically systolic group such that for every element g = 1 G of G if g n and g m are conjugated then n = m. Then the Conjugacy Problem is solvable for G.
Proof. Suppose G acts geometrically on a metrically systolic complex X. Let g = php −1 . By the assumption on conjugates of g, we may find n such that the displacement of g n is as large as in Lemma 3.5. Note that n does not depend on g, it only depends on the number of elements in the orbit of G contained in a ball of X of given size. Clearly g n = ph n p −1 . By Lemma 3.5 the displacement of p is bounded by value depending only on displacements of g, and h, and the action of G on X. Hence there is a bound on the number of possible p's. Note that this number is of the same order as the number of words we need to search in the CAT (0) case.  Case 1: S is not flag. Then we proceed exactly as in the proof of [Els09, Theorem 2.4]: we decompose S into two discs along an "empty" triangle, create two spheres of smaller area and use the induction assumption.
Case 2: S is flag. Since the 2-sphere does not admit a metric of non-positive curvature there exists a vertex v in S whose link, a cycle C, has angular length less than 2π. We have the decomposition S = D 1 ∪ D 2 , where D 1 is the star of v and D 2 is the complement of the interior of D 1 . By Lemma 2.5 the cycle f | C has a singular disc diagram D with no internal vertices. Let B 1 be the simplicial cone over D with apex v, and let F 1 : B 1 → X be the simplicial map such that F 1 (u) = f (u), for all vertices u (it is well defined by flagness of X). Then S 2 = D 2 ∪ D is a simplicial sphere of area smaller than the one of S. Let f 2 : S 2 → X be the simplicial map coinciding on vertices with f . Applying the inductive assumption we extend it to F 2 : B 2 → X, where B 2 is a triangulation of the ball with no internal vertices satisfying ∂B 2 = S 2 . Finally we put B = B 1 ∪ B 2 and F = F 1 ∪ F 2 .
Januszkiewicz-Światkowski introduced in [JŚ07] the notion of constant filling radius for k-spherical cycles, shortly S k FRC. This is a coarse feature of metric spaces saying, roughly, that in large scale every k-sphere has a filling within its uniform neighbourhood. A direct consequence of Theorem 3.7 is the following.
Corollary 3.8. Metrically systolic complexes and groups are S 2 FRC, that is, they have constant filling radius for 2-spherical cycles.
3.4. Morse Lemma for 2-dimensional quasi-discs. In this subsection we prove a Morse Lemma for 2-dimensional quasi-discs. It states, roughly speaking, that, for a given cycle C in a metrically systolic complex, a quasiisometrically embedded disc diagram is contained in an a-neighbourhood of any other singular disc diagram for C, with a independent of the size of the disc.
We use the combinatorial metric on simplicial complexes. In particular, the distance between adjacent vertices is 1. Let B(R, v) denote the (combinatorial) ball of radius R centered at v, that is the full subcomplex of a simplicial complex spanned by all vertices at distance at most R from v. Similarly, the sphere S(R, v) is the full subcomplex spanned by all vertices at distance R from v. Let T (r, R; v) denote the tube (annulus) of radii r, R around v, that is, the full subcomplex spanned by all vertices u such Recall that the systolic plane, denoted E 2 , is the triangulation of the Euclidean plane by regular triangles.
Theorem 3.9 (Morse Lemma for 2-dimensional quasi-discs). Let D be a combinatorial ball in the systolic plane E 2 . Let f : D → X be a disc diagram for a cycle C in X being an (L, A)-quasi-isometric embedding. Let g : D → X be a singular disc diagram for C. Then im(f ) ⊆ N a (im(g)), where a > 0 is a constant depending only on L and A.
Proof. There exist constants L ≥ L and A ≥ A depending only on L, A such that f : D → f (D) is an (L , A )-quasi-isometry, and there is an (L , A )quasi-isometryf : f (D) → D (0) such thatf • f and f •f are A -close to identities. Let K ≥ max{L , A , 3}. We will further work with K instead of L, A -this will make the computations easier. In particular (L , A )-quasiisometries are (K, K)-quasi-isometries. We claim that a = K 20 satisfies the assertion of the lemma.
To prove the claim suppose, by contradiction, that f (α) is null-homologous in T (K 8 , K 12 ; v) ∩ X 1 . Then there exists a simplicial map from a simplicial 2-complex T to T (K 8 , K 12 ; v) ∩ X 1 sending the boundary cycle to f (α). We define a mapf • h : T → D as follows. For every vertex u ∈ T we send it tof • h(u). An edge uw is sent to a geodesic betweenf (u) andf (w). A triangle uwz is sent to a singular disc in D bounded by the chosen geodesic between images of vertices. Sincē and since the image of every edge has diameter at most K, and similarly the image of every triangle has diameter at most K, we have that the image off • h is contained in T (K 6 − K, K 14 + K,f (v)). Furthermore, for every i, we have d(v i ,f (f (v i ))) ≤ K, and d(f (f (v i )),f (f (v i+1 ))) ≤ K. Therefore, there exists a homotopy between α and the image of f (α) byf • h within the 2K-neighborhood of α. It follows that α is null-homologous within T (K 5 , K 15 ;f (v)) -contradiction concluding the proof of the claim.
Let Y be a simplicial complex homeomorphic to an annulus (tube) in E 2 with the inner boundary cycle isomorphic to the boundary cycle C of D, and admitting a simplicial retraction on C. Observe that the boundary cycle of D is also C. Let D = D ∪ C Y be the complex obtained by gluing D and Y along C. Similarly, let D = D ∪ C Y . Both, D and D are non-singular discs, with isomorphic boundaries C -the other boundary cycle of Y . Consider a triangulated sphere S := D ∪ C D obtained by the identification of the boundaries, and the map ψ : S → X being the union of maps f , g, and the retraction maps sending copies of Y to their internal cycles C. By Theorem 3.7 there exists a simplicial extension of ψ to a three-ball without internal vertices. Hence [ψ] = 0 in H 2 (X 1 ∪ X 2 ; Z).
On the other hand the 1-cycle α is null-homotopic inside B(K 10 ,f (v)) ⊆ D. Hence there exists a simplicial disc D 1 ⊆ B(K 10 ,f (v)) providing the homotopy. Similarly, there is a disc D 2 ⊆ D − B(K 10 ,f (v)) ∪ C Y ∪ C D with boundary equal α. Observe that ψ(D 1 ) ⊆ X 1 , ψ(D 2 ) ⊆ X 2 , and ψ(α) ⊆ X 1 ∩ X 2 . Therefore, in the Mayer-Vietoris sequence for the pair X 1 , X 2 the boundary map Remark 3.10. In fact, a more general version of Lemma 3.9 could be proved following the same lines. Namely, we could require that f : D → X is a disc diagram being a quasi-isometry such that D is quasi-isometric to a ball in E 2 , rather than being the ball itself. Since the original statement allows technically much simpler proof, and it is the version that we subsequently use in [HO17b], we decided to formulate it this way.

The complexes for 2-generated Artin groups
In this section, we focus on 2-generated Artin groups. We construct metric simplicial complexes for them by modifying their Cayley complexes (see the "comments on the proof" subsection in the Introduction for an intuitive explanation). Later in Section 5 we will show these metric simplicial complexes are metrically systolic, and in Section 6 we will glue them together to form metrically systolic complexes for general two-dimensional Artin groups.
4.1. Precells in the presentation complex. Let DA n be the 2-generator Artin group presented by a, b | aba · · · n = bab · · · n . Let P n be the standard presentation complex for DA n . Namely the 1skeleton of P n is the wedge of two oriented circles, one labeled a and one labeled b. Then we attach the boundary of a closed 2-cell C to the 1skeleton with respect to the relator of DA n . Let C → P n be the attaching map. Let X * n be the universal cover of P n . Then any lift of the map C → P n to C → X * n is an embedding (cf. [HO17a, Corollary 3.3]). These embedded discs in X * n are called precells. Figure 6 depicts a precell Π * . X * n is a union of copies of Π * 's. We pull back the labeling and orientation of edges in P n to  Figure 6. Precell Π * .
obtain labeling and orientation of edges in X * n . We label the vertices of Π * as in Figure 6. The vertices and r are called the left tip and the right tip of Π * . The boundary ∂Π * is made of two paths. The one starting at , going along aba · · · n (resp. bab · · · n ), and ending at r is called the upper half (resp. lower half ) of ∂Π * . The orientation of edges inside one half is consistent, thus each half has an orientation. We summarize several basic properties of how these precells intersect each other. See [HO17a, Section 3.1] for proofs of these properties.
Lemma 4.1. Let Π * 1 and Π * 2 be two different precells in X * n . Then properly contained in the upper half or in the lower half of Π * 1 (and Π * 2 ); (3) if Π * 1 ∩ Π * 2 contains at least one edge, then one end point of Π * 1 ∩ Π * 2 is a tip of Π * 1 , and another end point of Π * 1 ∩ Π * 2 is a tip of Π * 2 , moreover, among these two tips, one is a left tip and one is a right tip.
Corollary 4.3. Let Π * 1 and Π * 2 be two different precells in X * n . If Π * 1 ∩ Π * 2 contains at least one edge, and Π * 3 ∩ Π * 2 = Π * 1 ∩ Π * 2 , then Π * 3 = Π * 1 . Proof. We apply Lemma 4.1 (3) to Π * 3 ∩ Π * 2 and Π * 1 ∩ Π * 2 to deduce that either Π * 1 and Π * 3 have the same left tip, or they have the same right tip. Thus Π * 1 = Π * 3 . 4.2. Subdividing and systolizing the presentation complex. We subdivide each precell in X * n as in Figure 7 to obtain a simplicial complex X n . A cell of X n is defined to be a subdivided precell, and we use the symbol Π for a cell. The original vertices of X * n in X n are called the real vertices, and the new vertices of X n after subdivision are called interior vertices. The interior vertex in a cell Π is denoted o as in Figure 7. (Here and further we use the convention that the real vertices are drawn as solid points and the interior vertices as circles.) Let Λ be the collection of all unordered pairs of cells of X n such that their intersection contains at least two edges (these intersections are connected by Lemma 4.1). For each (Π 1 , Π 2 ) ∈ Λ, we add an edge between the interior vertex of Π 1 and the interior vertex of Π 2 (cf. Figure 1). Denote the resulting complex by X n . It is clear that DA n acts on X n . Let X n be the flag completion of X n . Then X n is the simplicial complex we will work with. Now we give an alternative, but more detailed definition of X n . Pick a base cell Π in X n such that ∈ Π coincides with the identity element of DA n . Let Λ 0 be the collection of pairs of the form (Π, u −1 i Π), (Π, d −1 i Π) for i = 1, . . . , n − 2 (here each vertex of Π can be identified as an element of DA n , and u −1 i Π means the image of Π under the action of u −1 i ). Then the following is proved in [HO17a, Section 3.1].
(2) Different elements in Λ 0 are in different DA n -orbits. For each 1 ≤ i ≤ n − 2, we add an edge between o ∈ Π and u −1 i o ∈ u −1 i Π, and an edge between o ∈ Π and d −1 i o ∈ d −1 i Π. Then we use the action of DA n to add more edges in the equivariant way. The resulting complex is exactly X n , by Lemma 4.4.
Definition 4.5. We assign lengths to edges of X n . Edges between a real vertex and an interior vertex have length 1. Edges between two real vertices have length equal to the distance between two adjacent vertices in a regular (2n)-gon with radius 1. Now we assign lengths to edges between two interior vertices. First define a function φ : [0, π) → R as follows. Let ∆(ABC) be a Euclidean isosceles triangle with length of AB and AC equal to 1, and ∠ A (B, C) = α. Then φ(α) is defined to be the length of BC. For 1 ≤ i ≤ n − 2, let e i be the edge between o and u −1 i o (or o and d −1 i o). Then the length of e i is defined to be φ( i 2n 2π). Now we use the DA n action to define the length of edges between interior vertices in an equivariant way.
Note that Π ∩ u −1 i Π and Π ∩ d −1 i Π have n − i edges. Thus we have the following observation by using the DA n -action and Lemma 4.4.
Lemma 4.6. Suppose Π 1 ∩ Π 2 has m edges for m ≥ 2. Let o i ∈ Π i be the interior vertex for i = 1, 2. Then there is an edge between o i and o j in X n whose length is φ( n−m 2n 2π).
Lemma 4.7. The lengths of the three sides of each triangle in X (1) n satisfy the strict triangle inequality. Thus each 2-simplex of X n can be metrized as a non-degenerate Euclidean triangle whose three sides have length equal to the assigned length of the corresponding edges.
Proof. We only prove the case when this triangle is made of three interior vertices The other cases are already clear from the construction. By Lemma 4.2, Π 1 ∩ Π 2 and Π 1 ∩ Π 3 are contained in the same half (say upper half) of Π 1 , otherwise Π 2 ∩ Π 3 is at most one vertex, which contradicts that o 2 and o 3 are joined by an edge. We assume without loss of generality that Π 1 is the base cell Π. By Lemma 4.1 (3), each of Π 2 and Π 3 contains exactly one tip of Π 1 . We first consider the case when Π 2 contains the left tip of Π 1 and Π 3 contains the right tip of Π. Suppose Π 2 ∩ Π 1 (resp. Π 3 ∩ Π 1 ) contains m 2 (resp. m 3 ) edges. Then by Lemma 4.1 (3), Π 2 ∩ Π 3 contains m 2 + m 3 − n edges. By Lemma 4.6, length 2π). Note that π > n−(m 2 +m 3 −n) 2n 2π = n−m 2 2n 2π + n−m 3 2n 2π, thus we can place o 2 , o 1 , o 3 consecutively in the unit circle such that they span a Euclidean triangle with side lengths as required. Next we consider the case that both Π 2 and Π 3 contains the left tip of Π 1 . We assume without loss of generality that Π 1 ∩Π 2 Π 1 ∩Π 3 . Then, by Corollary 4.1 (3), the left tip of Π 3 is contained in Π 2 ∩ Π 3 . Thus we can repeat the argument in the previous case with Π 1 replaced by Π 3 . The case when both Π 2 and Π 3 contain the right tip of Π 1 can be handled similarly.
From now on, we think of each 2-simplex of X n as a Euclidean triangle with the required side lengths. If three vertices x 1 , x 2 and x 3 span a 2simplex in X n , then we use ∠ x 1 (x 2 , x 3 ) to denote the angle at x 1 of the associated Euclidean triangle.

The link of X n
In this section we study links of vertices in the complex X n defined in the previous section.
Choose a vertex v ∈ X n , let Λ v be the link lk(v, X (1) n ) of v in X n , i.e. Λ v is the full subgraph of X (1) n spanned by vertices which are adjacent to v. For an edge v 1 v 2 ⊂ Λ v , we define the angular length of this edge to be ∠ v (v 1 , v 2 ). This makes Λ v a metric graph. We define angular metric on Λ v in the same way as in Subsection 2.1 and use the notation from over there.
The main result of the section is the following proposition.
Proposition 5.1. Let v be a vertex of X n .
(1) The angular lengths of the three sides of each triangle in Λ v satisfy the triangle inequality.
We caution the reader that each edge in Λ v has an angular length, and has a length as defined in the previous section. Here we mostly work with angular length, but will switch to length occasionally. In this section we study the structure of Λ v with respect to the angular metric.
The proof of Proposition 5.1 is divided into two cases: the case of a real vertex v is treated in Subsection 5.1 and the case of an interior vertex v is treated in Subsection 5.2. In each case we first describe precisely the combinatorial and metric structure of the link and then we study in details angular lengths of simple cycles in the link. 5.1. Link of a real vertex. The main purpose of this subsection is to prove Proposition 5.1 for a real vertex v.
Since the links of any two real vertices are isomorphic as metric graphs with the angular metric, we can assume without loss of generality that v is the vertex l in the base cell Π (cf. Figure 7).
In the following proof, we will assume u 0 = d 0 = and u n = d n = r. Recall that each edge of X n which belongs to X * n has an orientation and is labeled by one of the generators a and b. We will first establish a sequence of lemmas towards the proof of Proposition 5.1.
The vertices of Λ v can be divided into two classes. (1) Edges between a real vertex and an interior vertex. These are exactly the edges of Λ v which are in X n , and they are called edges of type I. (2) Edges between two interior vertices. These are exactly the edges of Λ v which are not in X n , and they are called edges of type II. Note that there do not exist edges of Λ v which are between two real vertices. Now we characterize all edges of type I. See Figure 8 below for a picture of Λ v with only edges of type I shown.
(1) The collection of vertices in Λ v which are connected to b i (resp. a i ) by an edge of type I is exactly  i Π ∩ d −1 j Π equals to n − (|j − i|). We assume without loss of generality that i < j. Then the number of edges in d −1 There is an edge of type II between d −1 i o and d −1 j Π has at least two edges, thus (1) follows from the claim. (4) follows the claim and Lemma 4.6. (2) and (5) can be proved in a similar way. To see (3), note that d −1 i Π ∩ Π (resp. u −1 i Π ∩ Π) is contained in the upper half (resp. lower half) of Π. Thus (3) follows from Lemma 4.2.
(1) The angular lengths of the three sides of any triangle in Λ v satisfy the triangle inequality.
(2) Let ∆ be a 3-simplex in X n which contains a real vertex. Then there exists a (possibly degenerate) 3-simplex ∆ in the Euclidean 3-space 28 JINGYIN HUANG AND DAMIAN OSAJDA Figure 8. Edges of type I in the link of a real vertex such that there is a simplicial isomorphism ∆ → ∆ which preserves the lengths of edges.
Proof. Let ∆ be a triangle in Λ v . Since no two real vertices in Λ v are adjacent, ∆ either has two interior vertices, or three interior vertices. In the former case, since the angular length of any edge of type II is at most n−2 2n 2π (Lemma 5.3), it is less than the summation of the angular length of two edges of type I (Lemma 5.2), we consequently deduce that the triangle inequality holds. Moreover, (2) holds by triangle inequality and that the summation of the angular length of two edges of type I in ∆ is < π. In the latter case, by Lemma 5.3 (3), the three vertices of ∆ are either A similar equality holds with d replaced by u. Thus (1) and (2) follow.
We record a simple graph theoretic observation for later use.
Definition 5.5. A simple graph Γ is a tree of cliques if there are complete is a complete subgraph. Lemma 5.6. Let Γ be a tree of cliques. Then the following hold.
(1) Any simple n-cycle for n ≥ 4 in Γ is not 2-full.
(2) If Γ is a metric graph such that the three sides of each of its triangle satisfy the triangle inequality then, for any edge w 1 w 2 ⊂ Γ, the length of w 1 w 2 is bounded above by the length of any edge path from w 1 to w 2 .
Proof. For (1), we induct on the number k in Definition 5.5. Let ω ⊂ Γ be a simple n-cycle. If ω ⊂ ∪ k−1 i=1 ∆ i , then ω is not 2-full by induction. Now we assume ω ∪ k−1 i=1 ∆ i . Then there must be an edge e ⊂ ω such that e ∪ k−1 i=1 ∆ i . Let s, t be two vertices of e. By Definition 5.5 (1), e ⊂ ∆ k . Hence {s, t} ⊂ ∆ k . If {s, t} ⊂ ∪ k−1 i=1 ∆ i , then by Definition 5.5 (2) and the assumption that Γ is simple, Let t 1 and t 2 be two vertices in ω that are adjacent to s. Since n ≥ 4, t 1 and t 2 have combinatorial distance ≥ 2 in ω. By Definition 5.5 (1), the edge t 1 s is contained in one of the ∆ i . Thus we must have t 1 s ⊂ ∆ k . In particular t 1 ∈ ∆ k . Similarly, t 2 ∈ ∆ k . Thus there is an edge between t 1 and t 2 , and ω is not 2-full.
For (2), we can assume without loss of generality that w 1 w 2 together with another given edge path from w 1 to w 2 form a simple cycle. Thus it suffices to show that for any simple cycle ω ⊂ Γ, the length of an edge e ∈ ω is bounded above by the summation of the lengths of other edges in ω. Let n be the number of edges in ω. We induct on n. The case n = 3 follows from the assumption. The case n ≥ 4 follows from the induction assumption and from the fact that ω is not 2-full.
. . , u −1 n o}. Lemma 5.7. Each of Λ + v and Λ − v is a tree of cliques. Proof. We define the following sets of vertices of Λ + v .
Proof. It follows from Lemma 5.2 and Lemma 5.3 (3) that there are no edges between a vertex in Λ Since σ is a simple cycle, it follows that at least one of the following three situations happens: Thus the first statement follows. Lemma 5.7 and Lemma 5.6 imply that (1) and (2) are not possible, thus the second statement follows.
Lemma 5.9. Any edge path in Λ v from r −1 o to −1 o has angular length ≥ π.
Proof. Let ω be an edge path from We only prove the case ω ⊂ Λ + v since the other case is similar. Note that ω has to pass through at least one vertex in {d −1 i o} n−1 i=1 , so we can divide into the following four cases. Case 1: If there exists 1 < k < n − 1 such that d −1 k o ∈ ω, then Lemma 5.6 (2), Lemma 5.7 and Lemma 5.3 imply that length Then we must have b i ∈ ω (since there has to be a vertex in ω which is adjacent to This can be dealt in the same way as the previous case. Proof of Proposition 5.1 (for real vertices). Proposition 5.1 (1) follows from Corollary 5.4 and (2) follows from Lemma 5.8 and Lemma 5.9.
The following lemma will be used in Section 6.
Lemma 5.10. ( Recall that d ∠ denotes the angular metric on Λ v . Proof. Note that all edges of type II are between two interior vertices, and there are no edges between real vertices. Thus to travel from one real vertex to another real vertex in Λ v , one has to go through at least two edges of type I. Then (1) follows from Lemma 5.2 (3). Now we prove (2). Still, traveling from a i to a o has to go through at least two edges of type I. However, one readily verifies that only two edges of type I do not bring one from a i to a o . So we need at least one other edge. By Lemma 5.2 and Lemma 5.3, an edge in Λ v has angular length at least 1 2n 2π. Thus d ∠ (a i , a o ) ≥ π. On the other hand, the distance π can be realized by It is natural to ask when an edge path in Λ v from r −1 o to −1 o has angular length exactly = π. We record the following simple observation about such edge paths. The following will be crucial for applications in [HO17b].
Lemma 5.11. Suppose v is real and ω is an edge path in Λ v from r −1 o to −1 o of angular length π.
If ω ⊂ Λ + v , then the following are the only possibilities for ω: ( Note that ω is embedded, otherwise we can pass to a shorter path from r −1 o to −1 o, which contradicts Lemma 5.9. The statement ω ⊂ Λ + v or ω ⊂ Λ − v follows from the fact that there are no edges between a vertex in Λ + v \ {r −1 o, −1 o} and a vertex in Λ − v \ {r −1 o, −1 o}. Now we assume ω ⊂ Λ + v . If ω does not contain any real vertices, then we are in case (3), by Lemma 5.3 (4). If ω contains a real vertex, then it contains at least two edges of type I. Note that the angular length of ω with two edges of type I removed is π − n−1 2n 2π = 1 2n 2π, which equals to the smallest possible angular length of edges in Λ v . Thus we are in cases (1) or (2).

Link of an interior vertex.
In this subsection we prove Proposition 5.1 for an interior vertex v.
We assume without loss of generality that v is the interior vertex o of the base cell Π. Moreover, we assume v ∈ X n for n ≥ 3, since the n = 2 case is clear. Vertices of Λ v can be divided into the following two classes.
(2) Interior vertices. They are the interior vertices of some cell Π such that Π ∩ Π contains at least two edges. For the convenience of the proof, we name the vertices in ∂Π differently in this subsection. The vertices in the upper half (resp. lower half) of ∂Π are called v 0 , v 1 , . . . , v n (resp. v 0 , v 1 , . . . , v n ) from left to right. Note that v 0 = v 0 and v n = v n .
Let P be the collection of subcomplexes of ∂Π such that (1) they are homeomorphic to the unit interval [0, 1]; (2) each of them has m edges where 2 ≤ m ≤ n − 1; (3) each of them is contained in a half of ∂Π, and has nontrivial intersection with { , r} ⊂ ∂Π. By Lemma 4.1 (3), for each interior vertex of Λ v , the intersection of the cell containing this interior vertex and Π is an element in P. This actually induces a one to one correspondence between interior vertices of Λ v and elements of P by Corollary 4.3. Thus we can name the interior vertices of Λ v as follows. If the intersection of the cell which contains this interior vertex and Π is a path in the upper half (resp. lower half) of ∂Π that starts at and has i edges, then we denote this interior vertex by L i (resp. L i ). If the intersection of the cell which contains this interior vertex and Π is a path in the upper half (resp. lower half) of ∂Π that ends at r and has i edges, then we denote this interior vertex by R i (resp. R i ). Note that i is ranging from 2 to n − 1; see Figure 9. Let Π L i be the cell that contains L i . Figure 9. The link of an interior vertex Now we characterize edges in Λ v . They are divided into three classes.
(1) Edges of type I. They are edges between real vertices of Λ v . Hence they are exactly edges in ∂Π. Each of them has angular length = 1 2n 2π.
(2) Edges of type II. They are edges between a real vertex and an interior vertex, and they are characterized by Lemma 5.12 below.
(3) Edges of type III. They are edges between interior vertices of Λ v , and they are characterized by Lemma 5.13 below. We refer to Figure 9 for a picture of Λ v . Edges of type I and some edges of type II are drawn, but edges of type III are not drawn in the picture.
(1) The collection of vertices in ∂Π that are adjacent to L i (resp. L i ) is (2) The collection of vertices in ∂Π that are adjacent to R i (resp. R i ) is {v n , v n−1 , . . . , v n−i } (resp. {v n , v n−1 , . . . , v n−i }).
(3) The angular length of any edge between L i and a real vertex of Λ v is i 4n 2π. The same holds with L i replaced by L i , R i and R i . Proof. Note that {v 0 , v 1 , . . . , v i } are the vertices of ∂Π L i ∩ ∂Π. Thus the part of (1) concerning L i holds. We can prove the rest of (1), as well as (2), in a similar way.
is an isosceles triangle with v m being the apex, (3) follows.
(1) L i and L j (or R i and R j , L i and L j , R i and R j ) are connected by an edge in Λ v if and only if |j − i| ≤ n − 2. Moreover, the length of this edge is φ( |j−i| 2n 2π) (see Definition 4.5 for φ).
(2) L i and R j (or L i and R j ) are connected by an edge in Λ v if and only if i + j − n ≥ 2. Moreover, the length of this edge is φ( 2n−i−j 2n 2π).
(3) L i is not adjacent to any L j or R j . R i is not adjacent to any L j or R j .
Note that claims (1) and (2) concern the length, not the angular length of the edge.
Proof. We prove (1). Suppose without loss of generality that i < j. By Lemma 4.1 (3), the number of edges in Π L i ∩ Π L j is n − (j − i). Thus L i and L j are adjacent if and only if n − (j − i) ≥ 2. Now the length formula in (1) follows from Lemma 4.6. Other parts of (1) can be proved in a similar way.
(2) can be deduced in a similar way by noting that the number of edges in (3) follows from Lemma 4.2.
Corollary 5.14. The angular lengths of the three sides of each triangle in Λ v satisfy the triangle inequality.
Proof. The case when the triangle contains a real vertex follows from Corollary 5.4 (2) (consider the 3-simplex of X n spanned by this triangle and v). Now we assume the triangle has no real vertices.
Case 1: the three vertices of the triangle are L i , L j and L k with i < j < k. By Lemma 4.6, the length of oL i is φ( n−i 2n 2π). By Lemma 5.13 (1), the length of L i L j is φ( j−i 2n 2π). Since n−j 2n 2π + j−i 2n 2π = n−i 2n 2π, we can arrange L i , L j , L k , o in the unit circle as in Figure 10 left such that the distance between any two points in {L i , L j , L k , o} in the Euclidean plane equal to the length of the edge between them in X n . In particular, Case 2: the three vertices of the triangle are L i , L j and R k with i < j. By Lemma 5.13 (2), 2n−i−k 2n 2π < π and the length of L i R k is φ( 2n−i−k 2n 2π). Thus we can arrange L i , L j , o, R k as in Figure 10 right and argue as before. Then other cases are similar. Figure 10. Proof. We only consider Λ + v since Λ − v is similar. We define a sequence of collections of vertices of Λ + v as follows. Let S 1 = {L 2 , . . . , L n−1 }, S n−1 = {R n−1 , . . . , R 2 }, and S i = {L i+1 , . . . , L n−1 , R n−1 , . . . , R n−i+1 } for 2 ≤ i ≤ n−2. By Lemma 5.13, each S i spans a clique. Moreover, any pair of adjacent interior vertices in Λ v are contained in at least one of the S i . For . By Lemma 5.12, each V i spans a clique ∆ i . Moreover, any edge of Λ + v is contained in at least one of the Note that for each 2 ≤ i ≤ 2n − 2, ∆ i \ ∆ i−1 has exactly one vertex, and this vertex is not contained in ∪ i−1 m=1 ∆ m . Thus Definition 5.5 (2) holds, hence Λ + v is a tree of cliques.
Lemma 5.16. Let σ ⊂ Λ v be a simple cycle such that σ Λ + v and σ Λ − v . Then v 0 ∈ σ and v n ∈ σ. Consequently, if σ is a 2-full simple n-cycle in Λ v for n ≥ 4, then v 0 ∈ σ and v n ∈ σ.
Proof. It follows from Lemma 5.12 and Lemma 5.13 (3) that there are no edges between a vertex in Λ + v \{v 0 , v n } and a vertex in Λ − v \{v 0 , v n }. Based on Lemma 5.15, the rest of the proof is identical to the proof of Lemma 5.8.
Lemma 5.17. Any edge path in Λ v from v 0 to v n has angular length ≥ π.
Proof. Let ω ⊂ Λ v be an edge path from v 0 to v n . As in the proof of Lemma 5.9, we only consider the case ω ⊂ Λ + v . Case 1: There are two adjacent vertices of ω such that one is L j and another is R k . Note that v 0 is adjacent to L j , and R k is adjacent to v n . As in the proof of Lemma 5.14, we arrange L j , o and R k consecutively in a unit circle such that the Euclidean distance between any of two points in {L j , o, R k } equals to the length of the edge in X n between these two points.
Then L i is followed by a sub-path ω of ω with ω ⊂ ∂Π, and then a vertex R k . Suppose the first and the last vertices of ω are v m and v m respectively. Since L i and v m are adjacent, we have m ≤ i by Lemma 5.12 (1). Similarly, m ≥ n − j. By Lemma 5.6 (2) and Lemma 5.15, length We are done if i+j ≥ n. Now we assume i+j < n. Then m ≤ i < n−j ≤ m and length ∠ (ω ) ≥ n−j−i 2n 2π. Hence we still have length ∠ (ω) ≥ π. Case 3: Suppose case (1) is not true and ω ∩ {L 2 , . . . , L n−1 } = ∅. We suppose in addition that after the last vertex of ω in {L 2 , . . . , L n−1 } (say L i ), ω does not contain any vertex from {R 2 , . . . , R n−1 }.
Case 5: The remaining case is that ω does not contain any interior vertices. Then it is clear that length ∠ (ω) ≥ π.
Proof of Proposition 5.1 (for interior vertices). In view of Corollary 5.14, it suffices to prove any 2-full simplex n-cycle in Λ v with n ≥ 4 has angular length ≥ 2π. But this follows from Lemma 5.16 and Lemma 5.17.
The following is an analog of Lemma 5.11 in the case of interior vertex. It will be crucial for applications in [HO17b].
Lemma 5.18. Suppose v is interior and ω is an edge path in Λ v from v 0 to v n of angular length π.
If ω ⊂ Λ + v , then the following are the only possibilities for ω: (1) ω does not contain interior vertices, i.e. ω = v 0 → v 1 → · · · → v n ; Proof. We argue as Lemma 5.11 to show that ω is embedded, and that A left interior component of ω is a maximal connected sub-path of ω such that each of its vertices is one of the L i . We define a right interior component in a similar way. By case 1 of Lemma 5.17, there is at least one real vertex between a left interior component and a right interior component.
We show there is at most one left interior component. Suppose the contrary is true. Let L i be the first vertex of the last left interior component. The vertex in ω preceding L i is a real vertex, which we denote by v i 0 . Since ω is embedded, i 0 > 0. Let ω be the edge path consisting of the edge v 0 L i together with the part of ω from L i to v n . By Lemma 5.17, length ∠ (ω ) ≥ π.
which leads to a contradiction.
The same argument also shows that if L i ∈ ω then the vertex of ω preceding L i can not be some v i with i = 0. Thus, if there were a left interior component, then the vertex of ω following v 0 would be contained in such component.
Suppose there is a left interior component. Let L i be the last vertex in this component and let v i be any real vertex in the sub-path of ω from L i to v n . Then i ≥ i. To see this, we suppose the contrary i < i is true. Let ω be the edge path consisting of v 0 L i , L i v i , and the part of ω from v i to v n . By Lemma 5.6 (2) and Lemma 5.15, the angular length of the sub- Thus length ∠ (ω ) < length ∠ (ω) = π, which is contradictory to Lemma 5.17.
We claim that if there are two consecutive vertices L i and L j in ω such that ω reaches L i first, then i < j. To see this, note that by the proof of Corollary 5.14 (we can think the center of the circle in Figure 10 Thus if i > j, then the concatenation of v 0 L j and the sub-path of ω from L j to v n has angular length < length ∠ (ω) = π, which contradicts Lemma 5.17.
We can repeat the above discussion to obtain analogous statements for right interior components. Then the lemma follows.

The complexes for 2-dimensional Artin groups
In this section we finalize the proof of one of the main results of the article, namely Theorem 1.2 from Introduction. More precisely, for any two-dimensional Artin group A Γ we construct a metric simplicial complex X Γ , by gluing together complexes X n for 2-generated subgroups of A Γ . In Lemma 6.4 we prove that X Γ is simply connected, and in Lemma 6.6 we show that links of vertices in X Γ are 2π-large. As an immediate consequence we obtain the main result of this section: Theorem 6.1. X Γ is metrically systolic. Consequently, each 2-dimensional Artin group is metrically systolic.
Let A Γ be an Artin group with defining graph Γ. Let Γ ⊂ Γ be a full subgraph with induced edge labeling and let A Γ be the Artin group with defining graph Γ . Then there is a natural homomorphism A Γ → A Γ . By [vdL83], this homomorphism is injective. Subgroups of A Γ of the form A Γ are called standard subgroups.
Let P Γ be the standard presentation complex of A Γ , and let X * Γ be the universal cover of P Γ . We orient each edge in P Γ and label each edge in P Γ by a generator of A Γ . Thus edges of X * Γ have induced orientation and labeling. There is a natural embedding P Γ → P Γ . Since A Γ → A Γ is injective, P Γ → P Γ lifts to various embeddings X * Γ → X * Γ . Subcomplexes of X * Γ arising in such way are called standard subcomplexes. A block of X * Γ is a standard subcomplex which comes from an edge in Γ. This edge is called the defining edge of the block. Two blocks with the same defining edge are either disjoint, or identical.
We define precells of X * Γ as in Section 4.1, and subdivide each precell as in Figure 7 to obtain a simplicial complex X Γ . Interior vertices and real vertices of X Γ are defined in a similar way. We record the following simple observations. Lemma 6.2.
(1) Each element of A Γ maps one block of X Γ to another block with the same defining edge; (2) if g ∈ A Γ such that g maps an interior vertex of a block of X Γ to another interior vertex of the same block, then g stabilizes this block; (3) the stabilizer of each block of X Γ is a conjugate of a standard subgroup of A Γ .
Within each block of X Γ , we add edges between interior vertices as in Section 4.2. Then we take the flag completion to obtain X Γ . By Lemma 6.2, the newly added edges are compatible with the action of deck transformations A Γ X Γ . Thus the action A Γ X Γ extends to a simplicial action A Γ X Γ , which is proper and cocompact. A block in X Γ is defined to be the full subcomplex spanned by vertices in a block of X Γ . Two blocks of X Γ that have the same defining edge are either disjoint or identical. Lemma 6.3. Any isomorphism between a block in X * Γ and the space X * n (cf. Section 4.1) that preserves the labeling and orientation of edges extends to an isomorphism between a block in X Γ and the space X n (cf. Section 4.2).
Proof. By our construction, it suffices to show that if two vertices v 1 and v 2 in a block B ⊂ X * Γ are not adjacent in this block, then they are not adjacent in X * Γ . However, this follows from a result of Charney and Paris ( [CP14]) that B (1) is convex with respect to the path metric on the 1-skeleton of X * Γ . Lemma 6.4. X Γ is simply connected.
Proof. Let f be an edge of X Γ not in X Γ . Then there are two cells Π 1 and Π 2 such that f connects the interior vertices o 1 ∈ Π 1 and o 2 ∈ Π 2 . By construction, Π 1 and Π 2 are in the same block. Thus f and a vertex of Π 1 ∩ Π 2 span a triangle. By flagness of X Γ , f is homotopic rel its end points to the concatenation of other two sides of this triangle, which is inside X Γ . Now we show that each loop in X Γ is null-homotopic. Up to homotopy, we assume this loop is a concatenation of edges of X Γ . If some edges of this loop are not in X Γ , then we can homotop these edges rel their end points to paths in X Γ by the previous paragraph. Thus this loop is homotopic to a loop in X Γ , which must be null-homotopic since X Γ is simply connected.
Next, we assign lengths to edges of X Γ in an A Γ -invariant way. Let B ⊂ X Γ be a block with its defining edge labeled by n. By Lemma 6.3, there is a simplicial isomorphism i : B → X n that is label and orientation preserving. Note that all the edges between real vertices of X n has the same length, which we denote p. We define the length of an edge e ⊂ B to be length(i(e))/p. Then, for each vertex b ∈ B, the isomorphism lk(b, B (1) ) →lk(i(b), X (1) n ) induced by i preserves the angular lengths of edges. In particular, Proposition 5.1 holds for B.
We repeat this process for each block of X Γ . Each edge of X Γ belongs to at least one block, so it has been assigned at least one value of length. If an edge belongs to two different blocks, then both endpoints of this edge are real vertices, hence all values of lengths assigned to this edge equal to 1 by the previous paragraph. In summary, each edge of X Γ has a well-defined length. Moreover, such assignment of lengths is A Γ -invariant by Lemma 6.2. Lemma 6.5. Each simplex of X Γ is contained in a block.
Proof. Suppose there is an interior vertex v of the simplex ∆. Let Π be the cell containing v and B be the unique block containing v. Then any real vertex of X Γ adjacent to v is contained in Π and any interior vertex of X Γ adjacent to v is contained in B. Since B is a full subcomplex, we have ∆ ⊂ B. If ∆ does not contain any interior vertices, then ∆ is a point, or an edge, and the lemma is clear.
In particular, each triangle of X Γ is contained in a block, its side lengths satisfy the strict triangle inequality by Lemma 4.7. We define the angular metric on the link of each vertex of X Γ as before.
(1) The angular lengths of the three sides of each triangle in Λ v satisfy the triangle inequality.
Question 7.2. Let G be a metrically systolic group. Is the centralizer of any infinite order element in G abstractly commensurable with F k ×Z? Here F k is the free group with k generators and F 0 denotes the trivial group.
The answer is affirmative for systolic groups [OP18].
Question 7.4. Does every finite group acting on a metrically systolic complex fix a point?
A fixed point theorem for CAT(0) spaces follows from convexity of the distance function [BH99, Chapter II.2]. A fixed point theorem for systolic complexes has been proven in [CO15].
Question 7.5. Let X be a metrically systolic complex. Is X contractible? Does X satisfy S k F RC in the sense of [JŚ07] for all k ≥ 2?
It is proved in Section 3.3 that X has trivial second homotopy group and X is S 2 F RC. It is proved in [Che00,JŚ07] that the answer to Question 7.5 is affirmative for systolic complexes.
Question 7.6. Is there a notion of boundary for metrically systolic complexes which generalizes both the CAT (0) case and the systolic case?
See [BH99, Chapter II.8] for the definition of CAT (0) boundaries and [OP09] for the definition of boundaries of systolic complexes. 7.2. Abelian and solvable subgroups. For each Artin group A Γ , Charney and Davis [CD95b] defined an associated modified Deligne complex D Γ . Now we recall their construction in the 2-dimensional case. Vertices of D Γ are in one to one correspondence with left cosets of the form gA Γ , where g ∈ A Γ and Γ is either the empty-subgraph of Γ (in which case A Γ is the identity subgroup), or a vertex of Γ, or an edge of Γ. The rank of a vertex gA Γ is the number of vertices in Γ . Note that the set V of the vertices has a partial order induced by inclusion of sets. A collection form a chain with respect to the partial order. It is clear that D Γ is a 2-dimensional simplicial complex, and A Γ acts on D Γ without inversions, i.e. if an element of A Γ fixes a simplex of D Γ , then it fixes the simplex pointwise.
We endow D Γ with a piecewise Euclidean metric such that each triangle ∆(g 1 , g 2 A s , g 3 A st ) is a Euclidean triangle with angle π/2 at g 2 A s and angle π 2n at g 3 A st with n being the label of the edge st of Γ. By [CD95b, Proposition 4.4.5], D Γ is CAT (0) with such metric. As being observed in [Cri05, Lemma 6], the action A Γ D Γ is semisimple.
Theorem 7.7. Let A Γ be a 2-dimensional Artin group. Then every abelian subgroup of A Γ is quasi-isometrically embedded.
Proof. Let A ⊂ A Γ be an abelian subgroup. By [CD95b, Theorem B] and [CD95b, Corollary 1.4.2], the presentation complex of A Γ is a K(A Γ , 1) space. Thus A is a free abelian with rank ≤ 2. First we assume A ∼ = Z.
Since A Γ D Γ is semisimple, by [BH99, Chapter II.6], either A acts by translation on a CAT (0) geodesic line ⊂ D Γ , or A fixes a point x ∈ D Γ . In the former case, we conclude A is quasi-isometrically embedded by noting that any orbit map from A Γ to D Γ is coarsely Lipschitz. In the later case, since A Γ acts on D Γ without inversions, A fixes a vertex in D Γ . Thus, up to conjugation, we may assume that A is contained in a standard subgroup A Γ with Γ being a vertex or an edge. Any dihedral Artin group is CAT (0) [BM00], so A is quasi-isometrically embedded in A Γ (alternatively, dihedral Artin groups are C(4)-T(4) [Pri86], hence they are biautomatic [GS91]). By [CP14, Theorem 1.2], A Γ is quasi-isometrically embedded in A Γ . Hence A is quasi-isometrically embedded in A Γ . Now we assume A ∼ = Z ⊕ Z. By [BH99, Theorem II.7.20], either there is an A-invariant flat plane P ⊂ X Γ upon which A acts geometrically, or there is an A-invariant CAT (0) geodesic line ⊂ X Γ upon which A acts cocompactly, or A fixes a point. The first and the third case can be handled in a similar way. Now we assume the second case. There is a group homomorphism h : A → R by considering translation length of elements of A along . Since A acts on D Γ by cellular isometries, there exists > 0 such that any element of A with nonzero translation length has translation length > . Hence h(A) ∼ = Z. Thus by passing to a finite index subgroup of A, we assume A = a 1 , a 2 such that a 1 fixes a point in and a 2 has nonzero translation length.
Let p : A Γ → be the composition of an orbit map from A Γ to D Γ and the CAT (0) nearest point projection from D Γ to . Then there exists L > 0 such that p is L-Lipschitz. Suppose the translation length of a 2 is L . For i = 1, 2, suppose a i → A Γ is an L i -bi-Lipschitz embedding. For k = n 1 a 1 + n 2 a 2 ∈ A, let k ∞ = max{|n 1 |, |n 2 |}. Let * be the identity element in A Γ and let d A Γ denote the word metric on A Γ with respect to the standard generating set. If |n 1 | ≥ 2L 1 L 2 |n 2 |, then If |n 1 | < 2L 1 L 2 |n 2 |, then d A Γ (k, * ) ≥ L −1 d (p(n 1 a 1 + n 2 a 2 ), p( * )) = L −1 d (p(n 2 a 2 ), p( * )) ≥ L −1 L |n 2 | > L 2LL 1 L 2 k ∞ . Now we conclude from the above two inequalities that A → A Γ is a quasiisometric embedding.