{'fraction_non_alphanumeric': 0.04891535516801361, 'fraction_numerical': 0.019566142067205444, 'mean_word_length': 3.5565891472868216, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Two-Dimensional Systolic Complexes Satisfy Property A

We show that 2-dimensional systolic complexes are quasi-isometric to quadric complexes with flat intervals. We use this fact along with the weight function of Brodzki, Campbell, Guentner, Niblo and Wright to prove that 2-dimensional systolic complexes satisfy Property A.


Introduction
Property A is a quasi-isometry invariant of metric spaces introduced by Guoliang Yu in his study of the Baum-Connes conjecture [16]. It may be thought of as a non-equivariant version of amenability. As is the case for amenability, Property A has plenty of equivalent formulations. In particular, for finitely generated groups, it is equivalent to the exactness of the reduced C * -algebra of the group and also to the existence of an amenable action on a compact space [10,14]. Property A implies coarse embeddability into Hilbert space and hence the coarse Baum-Connes Conjecture and the Novikov Conjecture. Classes of groups for which Property A holds include Gromov hyperbolic groups [1], CAT(0) cubical groups [7], and uniform lattices in affine buildings [6], but it is an open question whether it holds for all CAT(0) groups.
In the current article we prove the following.
Main Theorem. Two-dimensional systolic complexes satisfy Property A.
A 2-dimensional systolic complex can be defined as a 2-dimensional simplicial complex which is CAT(0) when equipped with a metric in which each triangle is isometric to an equilateral Euclidean triangle. The class of isometry groups of such complexes is vast. It contains many Gromov hyperbolic groups. It also contains lattices inÃ 2 buildings, which were already proved to satisfy Property A by Campbell [6]. Some of these groups satisfy interesting additional properties such as Kazhdan's property (T). Notably, there are numerous well developed techniques for constructing groups acting on 2-dimensional systolic complexes (with various additional features) making them a rich source of examples. For instance, given a finite group F and a generating set S ⊆ F \ {1} whose Cayley graph Γ(F, S) has girth at least 6, Ballmann andŚwiatkowski [2, Theorem 2 and Section 4] construct a canonical infinite 2-dimensional systolic complex X whose oriented triangles are labeled by elements of S ∪ S −1 and in which the link of every vertex is isomorphic to Γ(F, S) with labels induced from the triangles. The labeled automorphisms of X act simply transitively on the oriented 1-simplices of X and if (F, S) has a relation of the form (st) 3 with s, t ∈ S then X has flat planes and so is not hyperbolic. This construction is a particular case of a development of a complex of groups as described in Bridson and Haefliger [4,Example 4.19(2) of Chapter III.C].
We prove the Main Theorem by showing that 2-dimensional systolic complexes are quasi-isometric to quadric complexes whose intervals with respect to a basepoint are CAT(0) square complexes (Theorem 3.1, Theorem 3.2 and Theorem 3.3). This allows us to apply the weight function and uniform convergence argument of Brodzki, Campbell, Guentner, Niblo and Wright [5] in their proof that finitely dimensional CAT(0) cube complexes satisfy Property A (Theorem 2.6).

Preliminaries
For basic algebraic topological notions such as those of CW complexes and simple connectedness we refer the reader to Hatcher [9]. A combinatorial map X → Y between CW complexes is one whose restriction to each cell of X is a homeomorphism onto a cell of Y . All graphs considered in this paper are simplicial. We consider metrics only on the 0-skeleton X 0 of a cell complex X as induced by the shortest path metric on the 1-skeleton X 1 . The metric ball B r (v) (respectively, metric sphere S r (v)) of radius r centered at a vertex v in a complex is the subgraph induced in the 1-skeleton by the set of vertices of distance at most (respectively, exactly) r from v. The girth of a graph is the length of the embedded shortest cycle. The link of a vertex v in a 2-dimensional simplicial complex is the graph whose vertices are the neighbours of v and where two vertices are joined by an edge if they span a triangle together with v. constants (C n ) n∈N and a family of functions f n,v : Γ 0 → N indexed by N × Γ 0 such that the following conditions hold. ( → 0 uniformly over all pairs of vertices (v, v ′ ) joined by an edge.
2.2. Two-dimensional systolic complexes. A 2-dimensional systolic complex is a simply connected 2-dimensional simplicial complex in which the girth of the link of every vertex is at least 6.
Then w and x are joined by an edge. (3) (Triangle Condition) Let u, v, w ∈ Y 0 such that v and w are joined by an edge and they are equidistant to u. Then there exists x ∈ Y 0 adjacent to v and w and closer to u than are v and w.

Quadric complexes.
A square complex is a 2-dimensional CW complex with a simplicial 1-skeleton such that the attaching maps of 2-cells are injective combinatorial maps from 4-cycles. The closed 2-cells of a square complex are called squares. We assume that no two squares of a square complex are glued to the same 4-cycle of the 1-skeleton. A quadric complex is a simply connected square complex in which, for any subcomplex on the left-hand side of Figure 1, there exists a replacement on the right-hand side. In other words, quadric complexes are simply connected generalized (4, 4)-complexes, as defined by Wise [15], that are built out of squares. They were first studied in depth by Hoda [11]. A quadric complex is a CAT(0) square complex if and only if its 1-skeleton is K 2,3 -free.

Lemma 2.3 (Quadrangle Condition)
. Let X be a quadric complex. Let u, v, w, x ∈ Y 0 such that v and w are adjacent to x and v and w are closer to u than is x. Then there exists y ∈ X 0 adjacent to v and w and closer to u than are v and w.
Lemma 2.3 follows from the fact that the 1-skeleta of quadric complexes are precisely the hereditary modular graphs. This characterization is due to the result of Hoda that the 1-skeleta are the bi-bridged graphs [11] and the theorem of Bandelt that a graph is bi-bridged iff it is hereditary modular [3,Theorem 1].
Let X be a quadric complex. The interval I(u, v) between a pair of vertices u and v in X 0 is the full subcomplex induced by the union of the geodesics between u and v.
Theorem 2.4. Let X be a quadric complex and let u, v ∈ X 0 . The 1-skeleton of I(u, v) is isometrically embedded in X 1 .
Proof. Suppose not. Then there are geodesics (α i ) ℓ i=0 and (β i ) ℓ i=0 from u to v and indices m and k such that no geodesic (γ i ) n i=0 from α m to β k is contained in I(u, v). Choose (α i ) i , (β i ) i , m and k so as to minimize n = d(α m , β k ). Without loss of generality, m ≤ k.
By Theorem 2.2 we may assume that (γ i ) i is contained in B k (u). Hence, since . But then appending γ n to (γ ′ i ) i we obtain a geodesic from α m to β k that is contained in I(u, v). This is a contradiction. □ Corollary 2.5. Intervals in quadric complexes are quadric.
Proof. Quadric complexes are characterized by metric properties of their 1-skeleta [11], so an isometrically embedded full subcomplex of a quadric complex is quadric. □ Fix a basepoint * ∈ X 0 . If, for all v ∈ X 0 , the interval I( * , v) is a CAT(0) square complex then (X, * ) has flat intervals. By Corollary 2.5, (X, * ) has flat intervals if and only if every I( * , v) is K 2,3 -free.
We now describe how we apply the results of Brodzki et al. [5] in the special case of 2-dimensional CAT(0) cube complexes to our present situation.
Let (X, * ) be a based quadric complex with flat intervals. Let Z v = I( * , v) be a CAT(0) square complex interval in a quadric complex. We will describe the weight function f n,v of Brodzki et al. [5] for v in Z v . For w ∈ Z 0 v , let ρ(w) be the number of neighbours of w in Z 1 v that lie on geodesics from w to * . The deficiency of w ∈ Z 0 v is defined as follows.
We extend f n,v by zeroes to all of Theorem 2.6. Let X be a quadric complex. If there exists * ∈ X 0 such that (X, * ) has flat intervals then X satisfies Property A.

The squaring of 2-dimensional systolic complexes
Let (Y, * ) be a 2-dimensional systolic complex with a basepoint * ∈ Y 0 . Let X 1 Y be the subgraph of Y 1 given by the union of all edges whose endpoints are not equidistant to u. Note that X 1 Y is bipartite The squaring of (Y, * ) is the based square complex (X Y , * ) obtained from X 1 Y by attaching a unique square along its boundary to each embedded 4-cycle of X 1 Y . Theorem 3.1. Let (Y, * ) be a based 2-dimensional systolic complex and let (X Y , * ) be the squaring of (Y, * ). Then Y 1 is quasi-isometric to X 1 Y . Proof. Applying Lemma 2.1(3) to * and an edge e of Y 1 that is not contained in X 1 Y gives us a triangle, one of whose edges is e and whose remaining edges are contained in X 1 Y . This ensures that distances in X 1 Y increase by at most a factor of two relative to distances in Y 1 .  Proof. We need to show that X Y is simply connected and that, for every subgraph of X 1 Y as in the left-hand side of Figure 1b, a pair of antipodal vertices in the outer 6-cycle is joined by an edge.
To show that X Y is simply connected it suffices to show that, for every embedded cycle α of X 1 Y , there is a 2-dimensional square complex D ′ homeomorphic to a 2-dimensional disk and a combinatorial map D ′ → X Y whose restriction to the boundary ∂D ′ of D ′ is α. By the van Kampen Lemma [13, Proposition 9.2 of Section III.9] since Y is simply connected, there exists a 2-dimensional simplicial complex D homeomorphic to a 2-disk D and a combinatorial map D → Y which restricts to α on the boundary. Choose such D → Y so as to minimize the number of triangles of D. By Lemma 2.1(1), each triangle of D has a unique edge e that is not contained in X 1 Y . Then e is contained in the interior of D and, by the minimality of D → Y , the star of e is embedded in Y . Let D ′ be the result of deleting all such e from D 1 and then spanning a square on each embedded 4-cycle. Since every embedded 4-cycle of X Y spans a square, we may extend (D ′ ) 1 → (X Y ) 1 to D ′ → X Y . This proves that X Y is simply connected.
Let W be a subgraph of X 1 Y as in the left-hand side of Figure 1b and with the same vertex labels. By 6-largeness of Y , each of the embedded 4-cycles of W have a diagonal. Since the girth of the link of u is at least 6, these diagonals must join u to each of the b i . Hence u is adjacent to every vertex in the outer 6-cycle C of W .
Let v be a furthest vertex of C from * . By Lemma 2.1(2), the neighbours of v in C are joined by an edge. But then there is a 5-cycle in the link of u which contradicts the 2-dimensional systolicity of Y . □ Consider the case where a 0 and a 1 are equidistant to * . Let a be the closer of * and v to a 0 and let b be the closer of * and v to b 0 . Let a ′ and b ′ be obtained by applying Lemma 2.3 to a 0 , a 1 and a and to b 0 , b 1 and b, as in Figure 3. By 6-largeness of Y , the 4-cycle (a 0 , a ′ , a 1 , b 0 ) has a diagonal in Y 1 . Since (a ′ , a 0 , b 0 ) is a geodesic, the diagonal must join a 0 and a 1 . Similarly, b 0 and b 1 are joined by edge in Y 1 and hence, by flagness, {a 0 , a 1 , b 0 , b 1 } spans a 3-simplex in Y . This contradicts the 2-dimensionality of Y .
In the remaining case a 0 and a 1 are not equidistant to * . Then the b i must all be equidistant to * with the (a 0 , b i , a 1 ) all geodesics. Applying a similar argument as in the previous case to the 4-cycles (a 0 , b i , a 1 , b j ) we see that the b i span a triangle in Y and so, together with a 0 , they span a 3-simplex contradicting, again, the 2-dimensionality of Y . □ As an immediate consequence of Theorem 3.1, Theorem 3.2, Theorem 3.3 and Theorem 2.6 we have the Main Theorem.