Square complexes and simplicial nonpositive curvature
SSquare complexes and simplicial nonpositive curvature
Tomasz Elsner a ∗ and Piotr Przytycki b † a Mathematical Institute, University of Wrocław,Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland e-mail: [email protected] b Institute of Mathematics, Polish Academy of Sciences,Śniadeckich 8, 00-956 Warsaw, Poland e-mail: [email protected]
Abstract
We prove that each nonpositively curved square VH -complex can beturned functorially into a locally -large simplicial complex of the samehomotopy type. It follows that any group acting geometrically on aCAT(0) square VH -complex is systolic. In particular the product oftwo finitely generated free groups is systolic, which answers a ques-tion of Daniel Wise. On the other hand, we exhibit an example of acompact non- VH nonpositively curved square complex, whose funda-mental group is neither systolic, nor even virtually systolic. In this note we compare nonpositively curved square VH -complexes (in-troduced in [Wis96]) and locally -large simplical complexes (introduced in[JŚ06]). First we describe locally -large and systolic complexes. The def-initions we use are taken from [JŚ07], with a slight modification allowingsimplices in a locally -large simplicial complex not to be embedded. Never-theless, the definition of a systolic complex coincides with the one in [JŚ07]. Definition 1.1. A generalised simplicial complex is a set S of affine simplicestogether with a set E (closed under compositions) of affine embeddings ofsimplices of S onto the faces of simplices of S (attaching maps), such thatfor any proper face τ of any simplex σ ∈ S there is precisely one attachingmap onto τ .A (generalised) simplicial map between generalised simplicial complexesis a set of affine maps commuting with the attaching maps and mapping eachsource simplex onto one of the target simplices. ∗ Partially supported by MNiSW grant N N201 541 738. † Partially supported by MNiSW grant N N201 541 738 and the Foundation for PolishScience. a r X i v : . [ m a t h . G R ] J u l he geometric realisation of a generalised simplicial complex ( S , E ) is thequotient space S / E . The quotient map of a generalised simplicial map isthe geometric realisation of such a map. We will abuse the language andnot distinguish between simplicial complexes or simplicial maps and theirgeometric realisations.The link of a simplex σ in a complex X = ( S , E ) is the (generalised)simplicial complex X σ = ( S σ , E σ ) where the set S σ is obtained by taking foreach attaching map φ σ,τ : σ → τ the maximal subsimplex of τ disjoint fromthe image of σ and E σ is the set of restrictions of the maps in E .Subsequently, we refer to a generalised simplicial complex simply as a simplicial complex and use the phrase simple simplicial complex when refer-ring to a standard simplicial complex (in which simplices are embedded andthe intersection of two simplices, if non-empty, is a single simplex). Definition 1.2.
A simplicial complex is simple if it does not contain an edgejoining a vertex to itself, or a pair of simplices with the same boundary (e.g.a double edge). A simple simplicial complex is flag if any complete subgraph(a clique) of its -skeleton spans a simplex.A cycle without diagonals in a simplicial complex X is an embeddedsimplicial loop such that there are no edges in X connecting a pair of itsnonconsecutive vertices.A simplicial complex is locally -large if all of its vertex links are flagand do not contain cycles of length or without diagonals. A connectedand simply connected locally -large simplicial complex is called systolic (i.e.systolic complexes are the universal coverings of connected locally -largecomplexes).A group admitting a geometric action on a systolic complex is called systolic .The original definition of local 6-largeness in [JŚ07] requires that we checkthe flagness and the absence of short cycles without diagonals for the linkat any simplex. However, for higher-dimensional simplices it is a directconsequence of those properties for the links at the vertices.Similarly as for simplicial complexes, we allow cells in square complexesnot to be embedded. The formal definition of a (generalised) square complexis the same as of a generalised simplicial complex, except for putting vertices,edges and squares in place of simplices . The only thing that needs to berephrased is the definition of the link. Definition 1.3.
The link at a vertex v of a (generalised) square complex X = ( S , E ) is a 1-dimensional (generalised) simplicial complex X v = ( S v , E v )
2a graph), where S v is obtained by taking for each attaching map φ v,σ : v → σ the vertex of σ opposite to v (if σ is an edge) or the diagonal of σ oppositeto v (if σ is a square) and E v is the set of restrictions of the maps in E .A square complex is called a VH -complex if its -cells can be partitionedinto two classes V and H called vertical and horizontal edges, respectively,and the attaching map of each square alternates between the edges of V and H . In other words, the link at each vertex is a bipartite graph withindependent sets of vertices coming from edges of V and H .Note that the link of a VH -complex at a vertex may have double edges. Definition 1.4.
A square complex is nonpositively curved (or locally CAT(0) )if the link at any vertex does not contain embedded combinatorial cycle oflength less than . For a VH complex this reduces to the property that thereare no double edges in the links at vertices.For a general definition of CAT(0) and nonpositively curved spaces (notneeded in our article) see [BH99]. Note only, that a simply connected spacewhich is nonpositively curved is CAT(0) ([BH99, Theorem 4.1]). Example 1.5.
The product of two trees is a CAT(0) VH -complex. If agroup acts freely by isometries on the product of two trees and preservesthe coordinates, then the quotient square complex is a nonpositively curved VH -complex.The paper is divided into two parts. In Section 2 we provide a functorialconstruction turning a nonpositively curved square VH -complex into a locally6-large simplicial complex of the same homotopy type (in particular turninga CAT(0) VH -complex into a systolic complex). The main application of theconstruction is: Theorem 1.6 (see Corollary 2.6) . The fundamental group of a compact non-positively curved VH -complex is systolic. The first application of Theorem 1.6 is the answer to a question posed byDaniel Wise in [Wis05]:
Corollary 1.7.
The product of two finitely generated free groups is systolic.
We also obtain a series of consequences of Theorem 1.6 by applying itto the examples of nonpositively curved VH -complexes (some with exoticproperties) given by Daniel Wise in [Wis96]. Corollary 1.8 (compare [Wis96, Corollary 2.8]) . The fundamental group ofan alternating knot complement is systolic. orollary 1.9 (compare [Wis96, Theorem 5.5]) . There exists a systolicgroup, which is not residually finite.
One can arrange for even a stronger property:
Corollary 1.10 (compare [Wis96, Theorem 5.13]) . There exists a systolicgroup, which has no finite-index subgroups.
In Section 3 we show that the VH -hypothesis in Theorem 1.6 is necessary: Theorem 1.11 (see Theorem 3.2) . There exists a compact non- VH nonpos-itively curved square complex, whose fundamental group is not systolic, noreven virtually systolic. Acknowledgements.
We thank Daniel Wise for motivating us, for hissuggestions and discussions.
V H -complexes are sys-tolic
Our main construction yields a way of turning a nonpositively curved VH -complex into a locally -large simplicial complex. Construction 2.1.
Let X be a VH complex with the sets E V and E H ofvertical and horizontal edges, respectively. Denote by V and S the sets ofvertices and squares of X , respectively. We construct an associated simplicialcomplex X ∗ called the simplexification of X , which has the same homotopytype as X .First we divide each vertical edge e ∈ E V in two and subdivide eachsquare s ∈ S into six triangles, as in the Figure 1(a), obtaining a (generalised)simplicial complex ˆ X (a triangulation of X ). The vertices of ˆ X (which willcorrespond to the vertices of X ∗ ) are in bijective correspondence with theelements of V ∪ E V ∪ S . We denote those vertices by v ∗ , e ∗ , s ∗ , for v ∈ V , e ∈ E V , s ∈ S , respectively.The link of ˆ X at a vertex e ∗ is isomorphic to the suspension of a set of n points, where n is the number of squares s ∈ S with a vertical edge e (counted with multiplicities, i.e. a square with both vertical edges equal to e is counted twice). The union ˆ Y e of all the simplices of ˆ X containing thevertex e ∗ is isomorphic to the suspension of an n -pod, where some pairs ofvertices may be identified.The complex X ∗ is obtained from ˆ X by attaching simplices σ + v = v ∗ + e ∗ s ∗ . . . s ∗ n and σ − v = v ∗− e ∗ s ∗ . . . s ∗ n for each vertex e ∗ , where v + and v − are the endpoints4igure 1: (a) the triangulation of X (b) ˆ Y e ⊂ ˆ X (c) Y ∗ e ⊂ X ∗ of the edge e ∈ E V and s , . . . , s n are the squares adjacent to the verticaledge e ∈ E V (counted with multiplicities).The link of e ∗ in X ∗ is the suspension of an ( n − -simplex and the union Y ∗ e of all the simplices of X ∗ containing the vertex e ∗ is isomorphic to thesuspension of an n -simplex, where some pairs of vertices may be identified. Proposition 2.2. A VH square complex X and its simplexification X ∗ havethe same homotopy type.Proof. As the triangulation ˆ X of X (defined in Construction 2.1) embedsinto X ∗ , we only need to prove that X ∗ deformation retracts onto ˆ X . Sincefor distinct e , e ∈ E V we have Y ∗ e ∩ Y ∗ e ⊂ ˆ X it is enough to show that forany e ∈ E V the complex Y ∗ e deformation retracts onto Y ∗ e ∩ ˆ X = ˆ Y e .If Y ∗ e is a simple complex (i.e. the suspension of the simplex with vertices s ∗ , . . . , s ∗ n , e ∗ ), then denoting by S the suspension and by C the cone operator,we have ˆ Y e = S ( C ( { s ∗ , . . . , s ∗ n } )) ⊂ S ( C ( σ ( s ∗ , . . . , s ∗ n ))) = Y ∗ e , where σ ( s ∗ , . . . , s ∗ n ) is the simplex with vertices s ∗ , . . . , s ∗ n .Consider the retraction r : C ( σ ( s ∗ , . . . , s ∗ n )) → C ( { s ∗ , . . . , s ∗ n } ) definedto be the affine extension of the map from the first barycentric subdivision,which preserves the subcomplex C ( { s ∗ , . . . , s ∗ n } ) and maps the barycentresof the remaining simplices to the cone vertex. It is easy to see that r canbe extended to a deformation retraction. By suspending the deformationretraction, we obtain a deformation retraction from Y ∗ e onto ˆ Y e .If Y ∗ e is not simple, then it is a quotient space of the suspension of asimplex obtained by identifying some pairs of vertices. In that case thedeformation retraction from Y ∗ e onto ˆ Y e is the quotient of the map describedabove. Remark 2.3.
Note that Construction 2.1 is functorial. Namely, let f : X → Y be a combinatorial map between VH complexes (i.e. mapping cells onto5ells of the same dimension, in our case mapping edges to edges and squaresto squares). Assume also that f preserves the sets of vertical and horizontaledges. Then f induces a canonical combinatorial map f ∗ : X ∗ → Y ∗ . More-over, we have ( f ◦ g ) ∗ = f ∗ ◦ g ∗ and id ∗ = id . In particular, if f is invertible(in other words is a combinatorial isomorphism ; it induces an isometry be-tween the geometric realisations), then so is f ∗ . Finally, note that if a group G acts properly (cocompactly, geometrically) on X , then its induced actionon X ∗ is also proper (cocompact, geometric).We are now ready for our main result. Theorem 2.4. If X is a nonpositively curved VH complex, then its simplex-ification X ∗ is locally -large. Before giving the proof, we list a few consequences, obtained by applyingProposition 2.2 and Remark 2.3.
Corollary 2.5. If X is a CAT(0) VH complex, then its simplexification X ∗ is systolic. If G acts geometrically on X , then G is systolic. There are two notable applications of Corollary 2.5.
Corollary 2.6 (Theorem 1.6) . The fundamental group of a compact non-positively curved VH complex is systolic. The second application promotes Wise’s aperiodic flat construction ([Wis96,Construction 7.1]) into the systolic setting.
Definition 2.7. A flat in a systolic complex X is a subcomplex E (cid:52) ⊂ X which is isomorphic to the equilaterally triangulated plane (the triangulationwith 6 triangles adjacent to each vertex) and whose 1-skeleton is isometricallyembedded into X (1) (with the combinatorial metric). Corollary 2.8.
There exists compact a locally 6-large simplicial complex,whose universal cover (which is systolic) contains a flat, which is not thelimit of a sequence of periodic flats.
It remains to prove our main result.
Proof of Theorem 2.4.
We need to check that the link of X ∗ at any vertexis flag and does not contain cycles of length 4 or 5 without diagonals. It isimmediate for any vertex e ∗ , where e ∈ E V , as the link of e ∗ is the suspensionof a simplex.The link of X at a vertex s ∗ , s ∈ S has the form of two suspensions ofsimplices, Sσ m − and Sσ n − ( m and n being the numbers of squares adjacent6igure 2: Sample link of X ∗ at a vertex (a) e ∗ (b) s ∗ (c) v ∗ to the vertical edges of s ), whose top and bottom vertices are connected byan edge (the case m = n = 3 is depicted in Figure 2(b)). It is clear that it isflag and any cycle without diagonals in that link has length at least 6.Now let L be the link of X ∗ at v ∗ , v ∈ V . Then L is the union of: • a set of simplices (one n e -simplex for each vertical edge e ∈ E V withan endpoint v , where n e is the number of squares s ∈ S adjacent to e ,counted with multiplicities) and • a set of m e -pods (one m e -pod for each horizontal edge e ∈ E H issuingfrom v , where m e is the number of squares adjacent to e , counted withmultiplicities),where each endpoint of each m e -pod is identified with a different vertex ofone of the simplices (Figure 2(c) shows the link L in the case when theneighbourhood of v ∈ V is the product of two tripods). It is clear that L isflag, and any cycle without diagonals in L needs to pass through at least two m e -pods and two simplices, which makes its length at least 6. In the next part of the paper we show that our theorem cannot be improvedto include all nonpositively curved square complexes. Namely, we constructan example of a compact nonpositively curved square complex, whose funda-mental group is not systolic. Later, we use that example to show a compactnonpositively curved square complex, whose fundamental group is neithersystolic, nor even virtually systolic (Theorem 1.11).Let K be the square complex presented in Figure 3, built up of twosquares. It has only one vertex and the link at this vertex is shown in Figure4. Thus we see that K is a nonpositively curved square complex, but not a VH -complex. We will show that π ( K ) is not a systolic group.7igure 3: The nonpositively curved square complex K Figure 4: The link of K at the only vertex Theorem 3.1.
The group π ( K ) is not systolic.Proof. The group π ( K ) = (cid:10) a, b, c | ba = ab − , a = cbc − (cid:11) is an HNN-extension of the fundamental group of a Klein bottle, so it hasa subgroup H = (cid:104) a, b (cid:105) , which is isomorphic to the fundamental group of aKlein bottle, in particular is virtually Z .Suppose π ( K ) is systolic, i.e. acts geometrically on some systolic simpli-cial complex X . As a corollary from the systolic flat torus theorem (preciselyby Corollary 6.2(1) together with Theorem 5.4 in [Els09]) we have that H ,as a virtually Z group, acts properly on a systolic flat in X (see Definition2.7). If the fundamental group of a Klein bottle (cid:104) a, b | ba = ab − (cid:105) acts prop-erly (by combinatorial isomorphisms) on an equilaterally triangulated plane E (cid:52) (shown in Figure 5), then the axis of the glide reflection a is l and thedirection of the translation b is k or vice versa.The elements a and b act by translations on E (cid:52) (with axes k and l ). The1-skeleton of E (cid:52) with the combinatorial metric is isometrically embeddedinto the 1-skeleton of X (by Definition 2.7), so the lines ˆ k and l (markedin Figure 5) are invariant geodesics (in the 1-skeleton) for a and b . By[Els10, Proposition 3.10] the geodesic l is quasi-convex in the 1-skeleton of X equipped with the combinatorial metric (i.e. any geodesic in X (1) with bothendpoints on l is contained in the δ -neighbourhood of l , for some universal δ ). The geodesic ˆ k is clearly not quasi-convex (every point of E (cid:52) lies on somegeodesic with both endpoints on ˆ k ). 8igure 5: E (cid:52) Since a = cb c − , the translation a has two invariant geodesics: ˆ k and c ( l ) (or l and c (ˆ k ) ). Two invariant geodesics of an isometry acting by atranslation on both of them are at finite Hausdorff distance, so either both ˆ k and c ( l ) are quasi-convex, or none of them is ([Els10, Proposition 3.11]).That contradicts the fact that l is quasi-convex, while ˆ k is not.As we have just shown, the fundamental group of K is not systolic, how-ever it is virtually systolic (there is a 2-leaf covering ˜ K , which is a VH -complex, so π ( ˜ K ) is systolic by Theorem 2.4). Now we use the complex K to construct a square complex S , whose fundamental group is not evenvirtually systolic.Let E be the compact nonpositively curved VH -complex which has noconnected finite coverings, constructed by Wise in [Wis96, Theorem 5.13].Let σ be any loop in E consisting entirely of horizontal edges. We can subdi-vide the complex K such that all loops a , b and c have the same combinatoriallength as the loop σ . Now we define ¯ E and ¯¯ E to be two copies of E and let S = ( E ∪ ¯ E ∪ ¯¯ E ) ∪ K/ ∼ , where ∼ is the identification of σ , ¯ σ and ¯¯ σ with a , b and c , respectively. Then S is a nonpositively curved (non- VH ) square complex. Theorem 3.2 (Theorem 1.11) . The group π ( S ) is not virtually systolic,where S is the nonpositively curved square complex defined above.Proof. We first argue that π ( S ) is not systolic. Since π ( S ) = π ( K ) ∗ a = σ π ( E ) ∗ b =¯ σ π ( ¯ E ) ∗ c =¯¯ σ π ( ¯¯ E ) is an amalgam product, the inclusion K ⊂ S induces injection π ( K ) → π ( S ) . To conclude that π ( S ) is not systolic, we can recall the fact that a9nitely presented subgroup of a torsion-free systolic group is systolic itself([Wis05]), while π ( K ) is not systolic (Theorem 3.1). An equivalent way ofarriving to that conclusion is to repeat for S the argument used for K in theproof of Theorem 3.1.To prove that π ( S ) is not virtually systolic, we show that it has no finite-index subgroups (i.e. S has no connected non-trivial finite coverings). Let p : ˜ S → S be a connected finite covering. Since E ⊂ S has no connectednon-trivial finite coverings, p − ( E ) is a disjoint union of copies of E . Inparticular, any lift ˜ a of the loop a has the same length as a . The same holdsfor the loops b and c . As a , b and c together with the three copies of π ( E ) generate π ( S ) , that implies that p is a trivial covering. References [BH99] M. R. Bridson and A. Haefliger,
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