Classifying spaces for families of subgroups of 8-located groups
aa r X i v : . [ m a t h . G R ] J a n CLASSIFYING SPACES FOR FAMILIES OF SUBGROUPS OF -LOCATED GROUPS IOANA-CLAUDIA LAZ ˘ARPOLITEHNICA UNIVERSITY OF TIMIS¸OARA, DEPT. OF MATHEMATICS,VICTORIEI SQUARE 2, 300006-TIMIS¸OARA, ROMANIAE-MAIL ADDRESS: [email protected]
Abstract.
We investigate the structure of the minimal displacement set in8-located complexes with the SD’-property. We show that such set embedsisometrically into the complex. Since 8-location and simple connectivity im-ply Gromov hyperbolicity, the minimal displacement set in such complex issystolic. Using these results, we construct a low-dimensional classifying spacefor the family of virtually cyclic subgroups of a group acting properly on an8-located complex with the SD’-property. Introduction
Curvature can be expressed both in metric and combinatorial terms. Metrically,one can refer to ’nonpositively curved’ (respectively, ’negatively curved’) metricspaces in the sense of Aleksandrov, i.e. by comparing small triangles in the spacewith triangles in the Euclidean plane (hyperbolic plane). These are the CAT(0)(respectively, CAT(-1)) spaces. Combinatorially, one looks for local combinatorialconditions implying some global features typical for nonpositively curved metricspaces.A very important combinatorial condition of this type was formulated by Gro-mov [10] for cubical complexes, i.e. cellular complexes with cells being cubes.Namely, simply connected cubical complexes with links (that can be thought assmall spheres around vertices) being flag (respectively, 5-large, i.e. flag-no-square)simplicial complexes carry a canonical CAT(0) (respectively, CAT(-1)) metric. An-other important local combinatorial condition is local k -largeness, introduced byJanuszkiewicz-´Swi¸atkowski [13] and Haglund [11]. A flag simplicial complex is locally k -large if its links do not contain ‘essential’ loops of length less than k .In particular, simply connected locally 7-large simplicial complexes, i.e. 7 -systolic complexes, are Gromov hyperbolic [13]. The theory of 7 -systolic groups , that isgroups acting geometrically on 7-systolic complexes, allowed to provide importantexamples of highly dimensional Gromov hyperbolic groups [12, 13, 22, 27, 29, 8].However, for groups acting geometrically on CAT(-1) cubical complexes or on7-systolic complexes, some very restrictive limitations are known. For example,7-systolic groups are in a sense ‘asymptotically hereditarily aspherical’, i.e. asymp-totically they can not contain essential spheres. This yields in particular that suchgroups are not fundamental groups of negatively curved manifolds of dimension Mathematics Subject Classification.
Primary 20F67, Secondary 05C99.
Key words and phrases. simplicial complex, 8-location, SD ′ property, Gromov hyperbolicity,minimal displacement set. above two; see e.g. [14, 20, 21, 27, 9, 25]. This rises need for other combinatorialconditions, not imposing restrictions as above. In [23, 5, 1, 4] some conditions ofthis type are studied – they form a way of unifying CAT(0) cubical and systolictheories.On the other hand, Osajda [24] introduced a local combinatorial condition of8 -location , and used it to provide a new solution to Thurston’s problem abouthyperbolicity of some 3-manifolds. In [16] we study of a version of 8-location,suggested in [24, Subsection 5.1]. This 8-location says that homotopically trivialloops of length at most 8 admit filling diagrams with one internal vertex. However,in the new 8-location essential 4-loops are allowed. In [16] (Theorem 4 .
3) it is shownthat simply connected, 8-located simplicial complexes are Gromov hyperbolic. Inthe current paper we give an application to this result.We focus on the study of the minimal displacement set in an 8-located complexsatisfying the SD ′ -property. One of the paper’s results states that such set isisometrically embedded into the complex. Moreover, we show that such set isGromov hyperbolic. In particular, it is systolic. This follows as an application ofthe fact that 8-located complexes with the SD ′ -property are Gromov hyperbolic(see [16]).For CAT(0) spaces and systolic complexes, however, studying the structure of theminimal displacement set is useful when constructing a low-dimensional classifyingspace for the family of virtually cyclic subgroups of a group acting properly on aCAT(0) space, respectively on a systolic complex (see [2], [26]). We expect similarresults in the 8-located case. Knowing that the minimal displacement set of an8-located complex with the SD ′ -property embeds isometrically into the complexand it is systolic, we will be able to apply results proven in[7] and [26] on systoliccomplexes. Acknowledgements . The author would like to thank Victor Chepoi, DamianOsajda and Tomasz Prytu la for useful discussions. This work was partially sup-ported by the grant 346300 for IMPAN from the Simons Foundation and the match-ing 2015 − Preliminaries
Let X be a simplicial complex. We denote by X ( k ) the k -skeleton of X, ≤ k < dim X . A subcomplex L in X is called full as a subcomplex of X if any simplex of X spanned by a set of vertices in L , is a simplex of L . For a set A = { v , ..., v k } ofvertices of X , by h A i or by h v , ..., v k i we denote the span of A , i.e. the smallest fullsubcomplex of X that contains A . We write v ∼ v ′ if h v, v ′ i ∈ X (it can happenthat v = v ′ ). We write v ≁ v ′ if h v, v ′ i / ∈ X . We call X flag if any finite set ofvertices which are pairwise connected by edges of X , spans a simplex of X .A cycle ( loop ) γ in X is a subcomplex of X isomorphic to a triangulation of S .A full cycle in X is a cycle that is full as a subcomplex of X . A k - wheel in X ( v ; v , ..., v k ) (where v i , i ∈ { , ..., k } are vertices of X ) is a subcomplex of X suchthat γ = ( v , ..., v k ) is a full cycle and v ∼ v , ..., v k . The length of γ (denoted by | γ | ) is the number of edges in γ . Given two cycles α, β of X , we denote by α ⋆ β their concatenation.We define the combinatorial metric on the 0-skeleton of X as the number ofedges in the shortest 1-skeleton path joining two given vertices. -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 3 A ball (sphere) B i ( v, X ) ( S i ( v, X )) of radius i around some vertex v is a fullsubcomplex of X spanned by vertices at combinatorial distance at most i (at com-binatorial distance i ) from v . Definition 2.1.
A simplicial complex is m - located , m ≥
4, if it is flag and everyfull homotopically trivial loop of length at most m is contained in a 1-ball.Let σ be a simplex of X . The link of X at σ , denoted by X σ , is the subcomplexof X consisting of all simplices of X which are disjoint from σ and which, togetherwith σ , span a simplex of X . We call a flag simplicial complex k -large if there areno full j -cycles in X , for j < k . We say X is locally k -large if all its links are k -large.We call X k -systolic if it is connected, simply connected and locally k -large . For k = 6, we abbreviate k -systolic to systolic.We introduce further a global combinatorial condition on a flag simplicial com-plex. Definition 2.2.
Let X be a flag simplicial complex. For a vertex O of X anda natural number n , we say that X satisfies the property SD ′ n ( O ) if for every i ∈ { , ..., n } we have:(1) (T) (triangle condition): for every edge e ∈ S i +1 ( O ), the intersection X e ∩ B i ( O ) is non-empty;(2) (V) (vertex condition): for every vertex v ∈ S i +1 ( O ), and for every twovertices u, w ∈ X v ∩ B i ( O ), there exists a vertex t ∈ X v ∩ B i ( O ) such that t ∼ u, w .We say X satisfies the property SD ′ ( O ) if SD ′ n ( O ) holds for each natural number n . We say X satisfies the property SD ′ if SD ′ n ( O ) holds for each natural number n and for each vertex O of X .The following result is given in [24]. Proposition 2.1.
A simplicial complex which satisfies the property SD ′ ( O ) forsome vertex O , is simply connected. Definition 2.3.
A group acting properly discontinously and cocompactly, by au-tomorphisms, on an m -located simplicial complex with the SD ′ property, is calledan m -located group, m ≥ Definition 2.4.
Given a path γ = ( v , v , ..., v n ) in a simplicial complex X , onecan tighten it to a full path γ ′ with the same endpoints by repeatedly applying thefollowing operations: • if v i and v j are adjacent in X for some j > i +1, then remove from the sequenceall v k where i < k < j ; • if v i and v j coincide in X for some j > i , then remove from the sequence all v k where i < k ≤ j .We call γ ′ a tightening of γ . We allow the trivial case when γ is already full.Then its tightening is the path itself.2.1. Minimal displacement set for CAT(0) spaces.
For CAT(0) spaces theminimal displacement set is studied in [2].
Definition 2.5.
Let X be a metric space and let h be an isometry of X . The dis-placement function of h is the function d h : X → R defined by d h ( x ) = d ( h ( x ) , x ).The translation length of h is the number | h | = inf { d h ( x ) | x ∈ X } . The set of points IOANA-CLAUDIA LAZ˘AR where d h attains this infimum is denoted by Min X (h) and it is called the minimaldisplacement set . Definition 2.6.
Let X be a metric space. An isometry h of X is called(1) elliptic if h has a fixed point,(2) hyperbolic if d h attains a strictly positive minimum.The following result concerns the structure of the minimal displacement set of ahyperbolic isometry h in a CAT(0) space. Theorem 2.2.
Let X be a CAT(0) space.(1) An isometry h of X is hyperbolic if and only if there exists a geodesic line c : R → X which is translated non-trivially by h ; namely h ( c ( t )) = c ( t + a ) ,for some a > . The set c ( R ) is called an axis of h . For any such axis, thenumber a is equal to | h | .Let h be a hyperbolic isometry of X .(2) The axes of h are parallel to each other and their union is Min X (h) .(3) Min X (h) is isometric to a product Y × R , and the restriction of h to Min X (h) is of the form ( y, t ) → ( y, t + | h | ) , where y ∈ Y , t ∈ R (see [2] , Theorem . , page ). Minimal displacement set for systolic complexes.
For systolic com-plexes the minimal displacement set is studied in [7].Let h be an isometry of a simplicial complex X . We define the displacementfunction d h : X (0) → N by d h ( x ) = d X ( h ( x ) , x ). The translation length of h isdefined as | h | = min x ∈ X (0) d h ( x ). If h does not fix any simplex of X , then h iscalled hyperbolic . In such case one has | h | >
0. Otherwise we call the isometry h elliptic . For a hyperbolic isometry h , we define the minimal displacement setMin X (h) as the subcomplex of X spanned by the set of vertices where d h attainsits minimum. Clearly Min X (h) is invariant under the action of h . Theorem 2.3.
Let h be a hyperbolic isometry of a systolic complex X . Then thesubcomplex Min X (h) is a systolic subcomplex, isometrically embedded into X (see [7] , Propositions . and . ). Let h be an isometry of a simplicial complex X . An h -invariant geodesic in X is called an axis of h . We say that Min X (h) is the union of axes, if for every vertex x ∈ Min X (h), there is an h -invariant geodesic passing through x , i.e. Min X (h) canbe written as follows:Min X (h) = span { S γ | γ is an h -invariant geodesic } (2 . h acts on X as a translation along the axes by the number | h | .For two subcomplexes X , X ⊂ X , the distance d min ( X , X ) is defined to be d min ( X , X ) = min { d X ( x , x ) | x ∈ X , x ∈ X } .Next we define the graph of axes denoted by Y h . For a hyperbolic isometry h satisfying (2 . Y h as follows: Y (0) h = { γ | γ is an h -invariant geodesic in Min X (h) } , Y (1) h = {{ γ , γ }| d min ( γ , γ ) ≤ } .Let d Y ( h ) denote the associated metric on Y (0) h . -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 5 Hyperbolicity.
One of the paper’s main results relies on the following theo-rem.
Theorem 2.4.
Let X be an -located simplicial complex which satisfies the SD ′ property. Then the -skeleton of X with a path metric induced from X (1) , is δ -hyperbolic, for a universal constant δ (see [16] , Theorem . ). We shall apply the following lemmas frequently.
Lemma 2.5.
Let X be an -located simplicial complex which satisfies the SD ′ n ( O ) property for some vertex O , n ≥ . Let v ∈ S n +1 ( O ) and let y, z ∈ B n ( O ) be suchthat v ∼ y, z and d ( y, z ) = 2 . Let w ∈ B n ( O ) be a vertex such that w ∼ y, v, z ,given by the vertex condition (V). We consider the vertices u , u ∈ B n − ( O ) suchthat u ∼ y, w and u ∼ w, z , given by the triangle condition (T). If u ≁ z and u ≁ y , then u ∼ u (see [16] , Lemma . ). Lemma 2.6.
Let X be an -located simplicial complex which satisfies the SD ′ n ( O ) property for some vertex O , n ≥ . Let v , v , v ∈ B n − ( O ) be such that v ∼ v ∼ v . Let w , w ∈ B n − ( O ) be such that w ∼ v , v and w ∼ v , v , given by thetriangle condition (T). Let p , p ∈ B n ( O ) be such that p ∼ v , v and p ∼ v , v ,given by the triangle condition (T). Then w ∼ w if and only if p ∼ p (see [16] ,Lemma . ). Classifying spaces with finite or virtually cyclic stabilisers.
The maingoal of this section is, given a group G , to describe a method of constructing amodel for a classifying space with virtually cyclic stabilisers out of a model for aclassifying space with finite stabilisers. The presented method is due to W. L¨uckand M. Weiermann ([19]). First we give the necessary definitions.A collection of subgroups F of a group G is called a family if it is closed undertaking subgroups and conjugation by elements of G . Two examples which will beof interest to us are the family FIN of all finite subgroups, and the family
VCY ofall virtually cyclic subgroups.
Definition 2.7.
Given a group G and a family of its subgroups F , a model for theclassifying space E F G is a G - CW -complex X such that for any subgroup H ⊂ G the fixed point set X H is contractible if H ∈ F , and empty otherwise.Let EG denote E FIN G and let EG denote E VCY G .A model for E F G exists for any group and any family. Any two models for E F G are G -homotopy equivalent (see [18]). However, general constructions alwaysproduce infinite dimensional models.We will describe a method of constructing a finite dimensional model for EG out of a model for EG and appropriate models associated to infinite virtually cyclicsubgroups of G . If H ⊂ G is a subgroup and F is a family of subgroups of G , let F ∩ H denote the family of all subgroups of H which belong to the family F . Moregenerally, if φ : H → G is a homomorphism, let φ ⋆ F denote the smallest family ofsubgroups of H that contains φ − ( F ) for all F ∈ F .Consider the collection VCY \ FIN of infinite virtually cyclic subgroups of G . Itis not a family since it does not contain the trivial subgroup. Define an equivalencerelation on VCY \ FIN by H ∼ H ⇐⇒ | H ∩ H | = ∞ IOANA-CLAUDIA LAZ˘AR
Let [ H ] denote the equivalence class of H , and let [ VCY ⊂ FIN ] denote the set ofequivalence classes. The group G acts on [ VCY ⊂ FIN ] by conjugation, and for aclass [ H ] ∈ [ VCY ⊂ FIN ] define the subgroup N G ( H ) ⊆ G to be the stabiliser of[ H ] under this action, i.e. N G ( H ) = { g ∈ G | | g − Hg ∩ H | = ∞} The subgroup N G ( H ) is called the commensurator of H , since its elements conju-gate H to the subgroup commensurable with H . For [ H ] ∈ [ VCY ⊂ FIN ] definethe family G [ H ] of subgroups of N G ( H ) as follows G [ H ] = { K ⊂ G | K ∈ [ VCY ⊂ FIN ] , [ K ] = [ H ] } ∪ { K ∈ FIN ∩ N G [ H ] } . Definition 2.8.
A group G satisfies condition (C) if for every g, h ∈ G with | h | = ∞ (infinite order) and any k, l ∈ Z we have gh k g − = h l = ⇒ | k | = | l | Lemma 2.7.
Let K ⊂ N G [ H ] be a finitely generated subgroup that contains somerepresentative of [ H ] and assume that the group G satisfies condition (C). Choosean element h ∈ H such that [ h h i ] = [ H ] (any element of infinite order has thisproperty). Then there exists k ≥ , such that h h k i is normal in K . For the proof see [26], Lemma 2 .
6, page 8.3.
Minimal displacement set for -located complexes with theSD’-property We study the structure of the minimal displacement set in an 8-located complexwith the SD’-property. The notations introduced in section 2 . Lemma 3.1.
Let h be a simplicial isometry without fixed points of a simplicialcomplex X . We choose a vertex v ∈ Min X (h) and a geodesic α ⊂ X (1) joining v with h ( v ) . Consider a simplicial path γ : R → X (where R is given a simplicial structurewith Z as the set of vertices) being the concatenation of geodesics h n ( α ) , n ∈ Z .Then γ is a | h | -geodesic (i.e. d ( γ ( a ) , γ ( b )) = | a − b | if a, b are such integers that | a − b | ≤ | h | ). In particular, Im( γ ) ⊂ Min X (h) .Proof. The proof is similar to the one given in [7], Fact 3 .
2. We prove the statementfor | a − b | = | h | (this implies the general case). Then, by the construction of γ ,either γ ( b ) = h ( γ ( a )) or γ ( a ) = h ( γ ( b )). Thus we have d ( γ ( a ) , γ ( b )) ≥ | h | . Theopposite inequality follows from the fact that γ is a simplicial map. (cid:3) Next we prove one of the paper’s main results.
Theorem 3.2.
Let h be a (simplicial) isometry with no fixed points of an -locatedcomplex X with the SD’-property. Assume | h | > . Then the -skeleton of Min X (h) is isometrically embedded into X .Proof. The construction is similar to the one given in [7], Proposition 3 . X (h) is not isometrically embedded. Then thereexist vertices v, w ∈ Min X (h) such that no geodesic in X with endpoints v and w iscontained in Min X (h). Choose v and w so that d ( v, w ) minimal (clearly d ( v, w ) > v with h ( v ), w with h ( w ) and v with w by geodesics α, β and γ , respectively. -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 7 Then h ( v ) is joined with h ( w ) by h ( γ ). Note that l ( α ) = l ( β ) = | h | , l ( γ ) = l ( h ( γ )) > α, β ⊂ Min X (h). Then, by minimality of d ( v, w ), geodesics α and γ intersect only at the endpoints. The same holds forthe geodesics α and h ( γ ), β and γ , β and h ( γ ), respectively. Suppose there is avertex x ∈ γ ∩ h ( γ ). Then h ( x ) ∈ h ( γ ) and h ( x ) = x , since h has no fixed points.We may assume, not losing generality, that h ( v ) , x, h ( x ) and h ( w ) lie on h ( γ ) inthis order. Then d ( x, h ( x )) = d ( h ( v ) , h ( x )) − d ( h ( v ) , x ) = d ( v, x ) − d ( h ( v ) , x ) ≤ d ( v, h ( v )) = | h | . So x ∈ Min X (h), contradicting the minimality of d ( v, w ). Thusthe geodesics α, β, γ, h ( γ ) either are pairwise disjoint but the endpoints or α and β have nonempty intersection. In both situations we proceed as follows.Let y, x be adjacent vertices on γ such that d ( y, v ) = d ( x, v ) −
1. It may happenthat y = v or x = w but not simultaneously due to the fact that d ( v, w ) > y is the last vertex of γ such that d ( y, h ( y )) = d ( y, v ) + d ( v, h ( v )) + d ( h ( v ) , h ( y )) (i.e. y is the last vertex of γ to be joined with h ( y ) by the leftof the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α ). The vertex x is the first vertex of γ such that d ( x, h ( x )) = d ( x, w ) + d ( w, h ( w )) + d ( h ( w ) , h ( x )) (i.e. x is the first vertex of γ tobe joined with h ( x ) by the right of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α ). Let v ′ ∈ γ, v ′ ∼ v (possibly with v ′ = y ).There are two cases: either l ( γ ) = 2 or l ( γ ) ≥ l ( γ ) = 2. Then y = x . Note that d ( v, h ( y )) = d ( w, h ( y )) = | h | + 1.Note that y ∈ B | h | ( h ( y )). Because v, w ∈ X y ∩ B | h | ( h ( y )), the (V) conditionof the SD’(h(y))-property implies that there exists a vertex t ∈ X y ∩ B | h | ( h ( y ))such that t ∼ v, w .Because v, t ∈ B | h | ( h ( y )), v ∼ t , the (E) condition of the SD’(h(y))-propertyimplies that there exists a vertex p ∈ B | h | ( h ( y )) such that p ∼ v, t .Because t, w ∈ B | h | ( h ( y )), t ∼ w , the (E) condition of the SD’(h(y))-propertyimplies that there exists a vertex q ∈ B | h | ( h ( y )) such that q ∼ t, w .Note that y ∈ B | h | ( h ( y )), v, t, w ∈ X y ∩ B | h | ( h ( y )), p, q ∈ B | h | ( h ( y )) , p ∼ v, t ; q ∼ t, w . Then Lemma 2.5 implies that p ∼ q .Let l ∈ β such that w ∼ l . Because q, l ∈ X w ∩ B | h | ( h ( y )), the (V) condition ofthe SD’(h(y))-property implies that there exists a vertex r ∈ X w ∩ B | h | ( h ( y )) suchthat r ∼ q, l .Because p, q ∈ B | h | ( h ( y )), p ∼ q , the (E) condition of the SD’(h(y))-propertyimplies that there exists a vertex m ∈ B | h |− ( h ( y )) such that m ∼ p, q .Because q, r ∈ B | h | ( h ( y )), q ∼ r , the (E) condition of the SD’(h(y))-propertyimplies that there exists a vertex n ∈ B | h |− ( h ( y )) such that n ∼ q, r .Note that t, w ∈ B | h | ( h ( y )), p, q, r ∈ B | h | ( h ( y )), m, n ∈ X q ∩ B | h |− ( h ( y )) , p, q ∈ X t , q, r ∈ X w . Then, because t ∼ w , Lemma 2.6 implies that m ∼ n .Let δ be the tightening of the cycle ( y, v, p, m, n, r, w ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f . Hence d ( y, m ) = 2. But y ∈ B | h | ( h ( y )) while m ∈ B | h |− ( h ( y )). Therefore d ( y, m ) = 3.This yields a contradiction.For the rest of the proof let l ( γ ) ≥ d ( v ′ , h ( v ′ )) = | h | + 2 or d ( v ′ , h ( v ′ )) = | h | + 1 or d ( v ′ , h ( v ′ )) = | h | .We analyze these cases below.Case A. Suppose d ( v ′ , h ( v ′ )) = | h | + 2. So there do not exist vertices a, b ∈ α such that v ′ ∼ a ∼ v , h ( v ′ ) ∼ b ∼ h ( v ). IOANA-CLAUDIA LAZ˘AR
Case A.1. Assume | γ | = 2 k, k ∈ N ⋆ .Assume w.l.o.g. d ( y, v ) = k . Then, due to the choice of the vertices x and y , wehave d ( x, w ) = k −
1. Recall y is the last vertex of γ to be joined with h ( y ) by theleft of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α ; x is the first vertex of γ to be joined with h ( x ) bythe right of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α .Let z ∈ γ such that z ∼ y , d ( z, v ) = d ( y, v ) −
1. Note that d ( z, h ( y )) = d ( x, h ( y )) = 2 k − | h | . Hence z, x ∈ X y ∩ B k − | h | ( h ( y )). Then the (V)condition of the SD’( h ( y ))-property implies that there exists a vertex t ∼ x, z suchthat t ∈ X y ∩ B k − | h | ( h ( y )).Note that z, t ∈ B k − | h | ( h ( y )) and z ∼ t . Then, by the (E) condition of theSD’( h ( y ))-property, there exists p ∈ B k − | h | ( h ( y )) such that p ∼ z, t .Note that t, x ∈ B k − | h | ( h ( y )) and t ∼ x . Then, by the (E) condition of theSD’( h ( y ))-property, there exists q ∈ B k − | h | ( h ( y )) such that q ∼ t, x .Note that y ∈ B k + | h | ( h ( y )), z, t, x ∈ X y ∩ B k − | h | ( h ( y )), p, q ∈ X t ∩ B k − | h | ( h ( y ), p ∼ z, q ∼ x . Then Lemma 2.5 implies that p ∼ q .If | γ | = 3, let u = w . If | γ | >
3, let u ∈ γ such that x ∼ u , d ( u, w ) = d ( x, w ) − d ( q, h ( y )) = d ( u, h ( y )) = 2 k − | h | . Hence q, u ∈ X x ∩ B k − | h | ( h ( y )).Then the (V) condition of the SD’( h ( y ))-property implies that there exists a vertex r ∼ q, u such that r ∈ X x ∩ B k − | h | ( h ( y )).Note that p, q ∈ B k − | h | ( h ( y )) and p ∼ q . Then by the (E) condition of theSD’( h ( y ))-property, there exists m ∈ B k − | h | ( h ( y )) such that m ∼ p, q .Note that q, r ∈ B k − | h | ( h ( y )) and q ∼ r . Then by the (E) condition of theSD’( h ( y ))-property, there exists n ∈ B k − | h | ( h ( y )) such that n ∼ q, r .Note that t, x ∈ B k − | h | ( h ( y )), p, q, r ∈ B k − | h | ( h ( y )), m, n ∈ X q ∩ B k − | h | ( h ( y )), p, q ∈ X t , q, r ∈ X x . Then, because t ∼ x , Lemma 2.6 implies that m ∼ n .Let δ be the tightening of the cycle ( y, z, p, m, n, r, x ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( y, m ) = 2. But y ∈ B k + | h | ( h ( y )) while y ∈ B k − | h | ( h ( y )). Therefore d ( y, m ) = 3. This yields a contradiction.Case A.2. Assume | γ | = 2 k + 1 , k ∈ N ⋆ .Assume first d ( v, y ) = k +1. Then d ( y, w ) = k . Note that d ( y, h ( y )) = 2 k +1+ | h | by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α and d ( y, h ( y )) = 2 k + | h | by the right ofthe cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . So the geodesic from y to h ( y ) passes by the right of thecycle γ ⋆ β ⋆ h ( γ ) ⋆ α . But the point y is chosen such that the geodesic from y to h ( y ) passes by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . The situation d ( v, y ) = k + 1is therefore not possible. So the only possible case is when d ( v, y ) = k . Therefore d ( y, w ) = k + 1, d ( x, w ) = k .Let z ∈ γ such that z ∼ y , d ( z, v ) = d ( y, v ) −
1. Note that d ( x, h ( x )) = d ( z, h ( x )) = 2 k + | h | . Because x, z ∈ X y ∩ B k + | h | ( h ( x )), the (V) condition of theSD’( h ( x ))-property implies that there exists a vertex t ∈ X y ∩ B k + | h | ( h ( x )) suchthat t ∼ x, z .Note that z, t ∈ B k + | h | ( h ( x )) and z ∼ t . Then, by the (E) condition of theSD’( h ( x ))-property, there exists p ∈ B k − | h | ( h ( y )) such that p ∼ z, t .Note that t, x ∈ B k + | h | ( h ( x )) and t ∼ x . Then, by the (E) condition of theSD’( h ( x ))-property, there exists q ∈ B k − | h | ( h ( x )) such that q ∼ t, x .Note that y ∈ B k +1+ | h | ( h ( x )), z, t, x ∈ X y ∩ B k + | h | ( h ( x )), p, q ∈ X t ∩ B k − | h | ( h ( x )) ,p ∼ z, q ∼ x . Then Lemma 2.5 implies that p ∼ q . -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 9 If | γ | >
5, let l ∈ γ such that z ∼ l , d ( l, v ) = d ( z, v ) −
1. If | γ | ∈ { , } , then l ∈ α such that z ∼ l , d ( l, v ) = d ( z, v ) −
1. Note that d ( l, h ( x )) = d ( p, h ( x )) = 2 k − | h | .Because l, p ∈ X z ∩ B k − | h | ( h ( x )), the (V) condition of the SD’( h ( x ))-propertyimplies that there exists a vertex s ∈ X z ∩ B k − | h | ( h ( x )) such that s ∼ l, p .Because l, s ∈ B k − | h | ( h ( x )) , l ∼ s , the (E) condition of the SD’( h ( x ))-propertyimplies that there exists a vertex m ∈ B k − | h | ( h ( x )) such that m ∼ l, s .Because s, p ∈ B k − | h | ( h ( x )) , s ∼ p , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex n ∈ B k − | h | ( h ( x )) such that n ∼ s, p .Because p, q ∈ B k − | h | ( h ( x )) , p ∼ q , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex r ∈ B k − | h | ( h ( x )) such that r ∼ p, q .Note that z ∈ B k + | h | ( h ( x )), l, s, p ∈ X z ∩ B k − | h | ( h ( x )), m, n ∈ X s ∩ B k − | h | ( h ( x )), m ∼ l, n ∼ p . Then Lemma 2.4 implies that m ∼ n .Note that z, t ∈ B k + | h | ( h ( x )), s, p, q ∈ B k − | h | ( h ( x )), n, r ∈ X p ∩ B k − | h | ( h ( x )), s, p ∈ X z , p, q ∈ X t , n ∼ s , r ∼ q . Then, because z ∼ t , Lemma 2.6 implies that n ∼ r .Let δ be the tightening of the cycle ( y, z, l, m, n, r, q, x ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( y, m ) = 2. But y ∈ B k +1+ | h | ( h ( y )) while m ∈ B k − | h | ( h ( y )). Therefore d ( y, m ) = 3. This yields a contradiction.In conclusion we have d ( v ′ , h ( v ′ )) = | h | + 2. This completes case A. Case B. There exists a vertex a ∈ α, v ∼ a ∼ v ′ . Suppose d ( v ′ , h ( v ′ )) = | h | + 1.Case B.1. Assume | γ | = 2 k, k ∈ N ⋆ .Assume w.l.o.g. d ( y, v ) = k . Then d ( y, h ( y )) = 2 k − | h | . Due to the choiceof the vertices x, y ∈ γ , we have d ( x, w ) = k −
1. Recall y is the last vertex of γ tobe joined with h ( y ) by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α ; x is the first vertex of γ to be joined with h ( x ) by the right of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α .Note that d ( y, h ( y )) = d ( x, h ( y )) = 2 k − | h | . Then y, x ∈ B k − | h | ( h ( y )).Because y ∼ x , the (E) condition of the SD’( h ( y ))-property implies that there existsa vertex t ∈ B k − | h | ( h ( y )) such that t ∼ y, x .Let l ∈ γ such that x ∼ l , d ( l, w ) = d ( x, w ) −
1. Note that t, l ∈ X x ∩ B k − | h | ( h ( y )). Then the (V) condition of the SD’( h ( y ))-property implies thatthere exists a vertex m ∈ X x ∩ B k − | h | ( h ( y )) such that m ∼ t, l .Because t, m ∈ B k − | h | ( h ( y )) , t ∼ m , the (E) condition of the SD’( h ( y ))-property implies that there exists a vertex r ∈ B k − | h | ( h ( y )) such that r ∼ t, m .Because m, l ∈ B k − | h | ( h ( y )) , m ∼ l , the (E) condition of the SD’( h ( y ))-property implies that there exists a vertex s ∈ B k − | h | ( h ( y )) such that s ∼ m, l .Note that x ∈ B k − | h | ( h ( y )), t, m, l ∈ X x ∩ B k − | h | ( h ( y )) and r, s ∈ X m ∩ B k − | h | ( h ( y )), r ∼ t, s ∼ l . Then Lemma 2.4 implies that r ∼ s .If | γ | = 4, then let u = w . If | γ | >
4, let u ∈ γ such that l ∼ u , d ( u, w ) = d ( l, w ) −
1. Note that s, u ∈ X l ∩ B k − | h | ( h ( y )). Then the (V) condition of theSD’( h ( y ))-property implies that there exists a vertex p ∈ X l ∩ B k − | h | ( h ( y )) suchthat p ∼ s, u .Because r, s ∈ B k − | h | ( h ( y )) , r ∼ s , the (E) condition of the SD’( h ( y ))-property implies that there exists a vertex c ∈ B k − | h | ( h ( y )) such that c ∼ r, s .Because s, p ∈ B k − | h | ( h ( y )) , s ∼ p , the (E) condition of the SD’( h ( y ))-property implies that there exists a vertex d ∈ B k − | h | ( h ( y )) such that d ∼ s, p .Note that m, l ∈ B k − | h | ( h ( y )), r, s, p ∈ B k − | h | ( h ( y )), c, d ∈ X s ∩ B k − | h | ( h ( y )), r, s ∈ X m , s, p ∈ X l . Then, because m ∼ l , Lemma 2.6 implies that c ∼ d . Let δ be the tightening of the cycle ( x, t, r, c, d, p, l ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( x, c ) = 2. But x ∈ B k − | h | ( h ( y )) while c ∈ B k − | h | ( h ( y )). Therefore d ( x, c ) = 3. This yields a contradiction.Case B.2. Assume | γ | = 2 k + 1 , k ∈ N ⋆ .Assume first d ( v, y ) = k +1. Then d ( y, w ) = k . Note that d ( y, h ( y )) = 2 k +1+ | h | by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α and d ( y, h ( y )) = 2 k + | h | by the right ofthe cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . So the geodesic from y to h ( y ) passes by the right of thecycle γ ⋆ β ⋆ h ( γ ) ⋆ α . But the point y is chosen such that the geodesic from y to h ( y ) passes by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . The situation d ( v, y ) = k + 1is therefore not possible. So the only possible case is when d ( v, y ) = k . Therefore d ( y, w ) = k + 1, d ( x, w ) = k .Note that d ( x, h ( x )) = d ( y, h ( x )) = 2 k + | h | . Hence y, x ∈ B k + | h | ( h ( x )). Then,since x ∼ y , the (E) condition of the SD’(h(x))-property implies that there exists avertex t ∈ B k − | h | ( h ( x )) such that t ∼ y, x .If | γ | = 3 , let z ∈ α, z ∼ y . If | γ | > , let z ∈ γ such that z ∼ y , d ( z, v ) = d ( y, v ) −
1. Note that z, t ∈ X y ∩ B k − | h | ( h ( x )). Then the (V) condition of theSD’( h ( x ))-property implies that there exists a vertex u ∈ X y ∩ B k − | h | ( h ( x )) suchthat u ∼ z, t .Because z, u ∈ B k − | h | ( h ( x )) , z ∼ u , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex p ∈ B k − | h | ( h ( x )) such that p ∼ z, u .Because u, t ∈ B k − | h | ( h ( x )) , u ∼ t , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex q ∈ B k − | h | ( h ( x )) such that q ∼ u, t .Note that y ∈ B k + | h | ( h ( x )), z, u, t ∈ X y ∩ B k − | h | ( h ( x )) and p, q ∈ X u ∩ B k − | h | ( h ( x )), p ∼ z, q ∼ t . Then Lemma 2.4 implies that p ∼ q .If | γ | = 3, let l ∈ α, l ∼ z . If | γ | = 5, let l = v . If | γ | >
5, let l ∈ γ such that l ∼ z , d ( l, v ) = d ( z, v ) −
1. Note that l, p ∈ X z ∩ B k − | h | ( h ( x )). Then the (V) conditionof the SD’( h ( x ))-property implies that there exists a vertex r ∈ X z ∩ B k − | h | ( h ( x ))such that r ∼ l, p .Because r, p ∈ B k − | h | ( h ( x )) , r ∼ p , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex n ∈ B k − | h | ( h ( x )) such that n ∼ r, p .Because p, q ∈ B k − | h | ( h ( x )) , p ∼ q , the (E) condition of the SD’( h ( x ))-property implies that there exists a vertex c ∈ B k − | h | ( h ( x )) such that c ∼ p, q .Note that z, u ∈ B k − | h | ( h ( x )), r, p, q ∈ B k − | h | ( h ( x )), r, p ∈ X z , p, q ∈ X u , n, c ∈ X p ∩ B k − | h | ( h ( x )). Then, because z ∼ u , Lemma 2.6 implies that n ∼ c .Let δ be the tightening of the cycle ( y, z, r, n, c, q, t ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( y, n ) = 2. But y ∈ B k + | h | ( h ( x )) while n ∈ B k − | h | ( h ( x )). Therefore d ( y, n ) = 3. This yields a contradiction.In conclusion we have d ( v ′ , h ( v ′ )) = | h | + 1. This completes case B. Case C. There exists a vertex b ∈ α such that h ( v ) ∼ b ∼ h ( v ′ ). Suppose d ( v ′ , h ( v ′ )) = | h | + 1.Case C.
1. Assume | γ | = 2 k, k ∈ N ⋆ . Assume w.l.o.g. d ( y, v ) = k . Then, dueto the choice of the vertices y, x ∈ γ , we have d ( x, w ) = k −
1. Recall y is the lastvertex of γ to be joined with h ( y ) by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α ; x is thefirst vertex of γ to be joined with h ( x ) by the right of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 11 Note that d ( h ( y ) , y ) = d ( h ( x ) , y ) = 2 k − | h | . Because h ( y ) , h ( x ) ∈ B k − | h | ( y )and h ( y ) ∼ h ( x ), the (E) condition of the SD’( y )-property implies that there existsa vertex t ∈ B k − | h | ( y ) such that t ∼ h ( y ) , h ( x ).If | γ | = 3, let l = h ( w ). If | γ | >
3, let l ∈ h ( γ ) such that l ∼ h ( x ), d ( l, h ( w )) = d ( h ( x ) , h ( w )) −
1. Note that t, l ∈ X h ( x ) ∩ B k − | h | ( y ). Then the (V) conditionof the SD’( y )-property implies that there exists a vertex u ∈ X h ( x ) ∩ B k − | h | ( y )such that u ∼ t, l .Note that t, u ∈ B k − | h | ( y ) , t ∼ u , the (E) condition of the SD’( y )-propertyimplies that there exists a vertex p ∈ B k − | h | ( y ) such that p ∼ t, u .Note that u, l ∈ B k − | h | ( y ) , u ∼ l , the (E) condition of the SD’( y )-propertyimplies that there exists a vertex q ∈ B k − | h | ( y ) such that q ∼ u, l .Note that h ( x ) ∈ B k − | h | ( y ), t, u, l ∈ X h ( x ) ∩ B k − | h | ( y ) and p, q ∈ X u ∩ B k − | h | ( y ), p ∼ t, q ∼ l . Then Lemma 2.5 implies that p ∼ q .If | γ | = 4, then z ∈ β , z ∼ l . If | γ | = 6, then z = w . If | γ | >
6, let z ∈ h ( γ )such that z ∼ l , d ( z, h ( w )) = d ( l, h ( w )) −
1. Note that q, z ∈ X l ∩ B k − | h | ( y ).Then the (V) condition of the SD’( y )-property implies that there exists a vertex n ∈ X l ∩ B k − | h | ( y ) such that n ∼ q, z .Because p, q ∈ B k − | h | ( y ) , p ∼ q , the (E) condition of the SD’( y )-propertyimplies that there exists a vertex r ∈ B k − | h | ( y ) such that r ∼ p, q .Because q, n ∈ B k − | h | ( y ) , q ∼ n , the (E) condition of the SD’( y )-propertyimplies that there exists a vertex c ∈ B k − | h | ( y ) such that c ∼ q, n .Note that u, l ∈ B k − | h | ( y ), p, q, n ∈ B k − | h | ( y ), p, q ∈ X u , q, n ∈ X l , r, c ∈ X q ∩ B k − | h | ( y ) , p ∼ r, n ∼ c . Then, because u ∼ l , Lemma 2.6 implies that r ∼ c .Let δ be the tightening of the cycle ( h ( x ) , t, p, r, c, n, l ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( h ( x ) , r ) = 2. But h ( x ) ∈ B k − | h | ( y ) while r ∈ B k − | h | ( y ). Therefore d ( h ( x ) , r ) = 3. This yields a contradiction.Case C.
2. Assume | γ | = 2 k + 1 , k ∈ N ⋆ .Assume first d ( v, y ) = k +1. Then d ( y, w ) = k . Note that d ( y, h ( y )) = 2 k +1+ | h | by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α and d ( y, h ( y )) = 2 k + | h | by the right ofthe cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . So the geodesic from y to h ( y ) passes by the right of thecycle γ ⋆ β ⋆ h ( γ ) ⋆ α . But the point y is chosen such that the geodesic from y to h ( y ) passes by the left of the cycle γ ⋆ β ⋆ h ( γ ) ⋆ α . The situation d ( v, y ) = k + 1is therefore not possible. So the only possible case is when d ( v, y ) = k . Therefore d ( y, w ) = k + 1, d ( x, w ) = k .Note that d ( h ( x ) , x ) = d ( h ( y ) , x ) = 2 k + | h | . Because h ( y ) , h ( x ) ∈ B k + | h | ( x )and h ( y ) ∼ h ( x ), the (E) condition of the SD’( x )-property implies that there existsa vertex t ∈ B k − | h | ( x ) such that t ∼ h ( y ) , h ( x ).If | γ | = 3, let l = h ( w ). If | γ | >
3, let l ∈ h ( γ ) such that l ∼ h ( x ), d ( l, h ( w )) = d ( h ( x ) , h ( w )) −
1. Note that t, l ∈ X h ( x ) ∩ B k − | h | ( x ). Then the (V) conditionof the SD’( x )-property implies that there exists a vertex u ∈ X h ( x ) ∩ B k − | h | ( x )such that u ∼ t, l .Because t, u ∈ B k − | h | ( x ) , t ∼ u , the (E) condition of the SD’( x )-propertyimplies that there exists a vertex p ∈ B k − | h | ( x ) such that p ∼ t, u .Because u, l ∈ B k − | h | ( x ) , u ∼ l , the (E) condition of the SD’( x )-propertyimplies that there exists a vertex q ∈ B k − | h | ( x ) such that q ∼ u, l . Note that h ( x ) ∈ B k + | h | ( x ), t, u, l ∈ X h ( x ) ∩ B k − | h | ( x ), p, q ∈ X u ∩ B k − | h | ( x ), p ∼ t, q ∼ l . Then Lemma 2.5 implies that p ∼ q .If | γ | = 3, let z ∈ β . If | γ | = 5, let z = h ( w ). If | γ | >
5, let z ∈ h ( γ )such that z ∼ l , d ( z, h ( w )) = d ( l, h ( w )) −
1. Note that q, z ∈ X l ∩ B k − | h | ( x ).Then the (V) condition of the SD’( x )-property implies that there exists a vertex s ∈ X l ∩ B k − | h | ( x ) such that s ∼ q, z .Because p, q ∈ B k − | h | ( x ) , p ∼ q , the (E) condition of the SD’( x )-propertyimplies that there exists a vertex c ∈ B k − | h | ( x ) such that c ∼ p, q .Because q, s ∈ B k − | h | ( x ) , q ∼ s , the (E) condition of the SD’( x )-propertyimplies that there exists a vertex d ∈ B k − | h | ( x ) such that d ∼ q, s .Note that u, l ∈ B k − | h | ( x ), p, q, s ∈ B k − | h | ( x ), p, q ∈ X u , q, s ∈ X l , c, d ∈ X q ∩ B k − | h | ( y ). Then, because u ∼ l , Lemma 2.6 implies that c ∼ d .Let δ be the tightening of the cycle ( h ( x ) , t, p, c, d, s, l ). Note that | δ | ≤ δ is full. Then, by 8-location, there is a vertex f such that δ ⊂ X f .Hence d ( h ( x ) , c ) = 2. But h ( x ) ∈ B k + | h | ( x ) while c ∈ B k − | h | ( x ). Therefore d ( h ( x ) , c ) = 3. This yields a contradiction.In conclusion we have d ( v ′ , h ( v ′ )) = | h | + 1. This completes case C. Case D. There exist vertices a, b ∈ α such that v ′ ∼ a ∼ v , h ( v ′ ) ∼ b ∼ h ( v ).Then d ( v ′ , h ( v ′ )) = | h | which yields a contradiction. (cid:3) Lemma 3.3.
Let h be a (simplicial) isometry with no fixed points of an -locatedcomplex X with the SD’-property. Let Y = Min X (h) . Then Y = Min Y (h) .Proof. Let x ∈ X such that d X ( x, h ( x )) = | h | . Then x ∈ Y . Let y = h ( x ) ∈ Y such that d Y ( y, h ( y )) = | h | . So y ∈ Min Y (h). Since d X ( x, h ( x )) = d Y ( y, h ( y )), wehave Y = Min Y (h). (cid:3) The construction of a low-dimensional classifying space for the family of virtuallycyclic subgroups of a group acting properly on an 8-located complex with the SD ′ -property relies on the following result. Theorem 3.4.
Let h be a (simplicial) isometry having no fixed points with | h | > ,of an -located complex X with the SD’-property. Then the set Min X (h) is Gromovhyperbolic. In particular, Min X (h) is systolic.Proof. Theorem 2.4 implies that X is Gromov hyperbolic. Let Y = Min X (h).Lemma 3.3 implies that Y = Min Y (h). The proof is by contradiction. Supposethere exists a k -wheel γ = ( z ; x , ..., x k ) ⊂ Y , 5 ≤ k ≤
6. According to Lemma3.2, the 1-skeleton of Y is isometrically embedded into X . Then the k -wheel γ alsobelongs to X . Due to the Gromov hyperbolicity of X , this yields a contradiction.So there does not exist any k -wheel in Y , 5 ≤ k ≤
6. This implies that Y is Gromovhyperbolic. In particular, Y is systolic. (cid:3) The following results on 8-located complexes with the SD’-property are imme-diate consequences of the fact that the minimal displacement set of a nonellipticisometry acting on such complex is a systolic subcomplex and it embeds isomet-rically into the complex. Their systolic analogues, also given below, imply thesesimilarities. We shall refer to these results when constructing a low-dimensional -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 13 classifying space for the family of virtually cyclic subgroups of a group acting prop-erly on an 8-located complex with the SD’-property.
Theorem 3.5.
Let h be a nonelliptic simplicial isometry of a uniformly locallyfinite systolic complex X . Then there is an h n -invariant geodesic for some n ≥ . For the proof see [7], Theorem 3 .
5, page 46.
Theorem 3.6.
Let h be a nonelliptic simplicial isometry of a uniformly locallyfinite -located complex X with the SD’-property. Assume | h | > . Then in X there is an h n -invariant geodesic for some n ≥ .Proof. Let Y = Min X (h). Since | h | >
3, Theorem 3.4 implies that Y is systolic.Then, by Theorem 3.5, there is in Y an h n -invariant geodesic γ for some n ≥ Y (1) is isometrically embedded into X , the h n -invariantgeodesic γ also belongs to X . This completes the proof. (cid:3) Theorem 3.7.
Let h be a simplicial isometry of a uniformly locally finite systoliccomplex X . Then either there is an h -invariant simplex (elliptic case) or there isan h -invariant thick geodesic (hyperbolic case). For the proof see [7], Theorem 3 .
8, page 49.
Theorem 3.8.
Let h be a simplicial isometry of a uniformly locally finite -locatedcomplex X with the SD’-property. Assume | h | > . Then either there is an h -invariant simplex (elliptic case) or there is an h -invariant thick geodesic (hyperboliccase).Proof. Let Y = Min X (h). Since | h | >
3, Theorem 3.4 implies that Y is systolic.Then, by Theorem 3.7, in Y either there is an h -invariant simplex (elliptic case)or there is an h -invariant thick geodesic (hyperbolic case). Since, by Theorem3.2, Y (1) is isometrically embedded into X , this h -invariant simplex (elliptic case),respectively this h -invariant thick geodesic (hyperbolic case) also belongs to X . (cid:3) Theorem 3.9.
Let h be a nonelliptic simplicial isometry of a uniformly locallyfinite systolic complex X . If there exists an h n -invariant geodesic in X , then forany vertex x ∈ Min X (h n ) ⊂ X , there exists an h n -invariant geodesic passing through x . For the proof see [7], Remark page 48.
Theorem 3.10.
Let h be a nonelliptic simplicial isometry of a uniformly locallyfinite -located complex X with the SD’-property. Assume | h | > . If there existsan h n -invariant geodesic in X , then for any vertex x ∈ Min X (h n ) ⊂ X , there existsan h n -invariant geodesic passing through x .Proof. Let Y = Min X (h). Theorem 3.4 implies that Y is systolic. According toTheorem 3.5, in Y (and then, by Theorem 3 .
6, also in X ) there exists an h n -invariant geodesic for some n ≥
1. Hence, by Theorem 3.9, for any vertex x ∈ Min X (h n ) ⊂ Y, there exists an h n -invariant geodesic passing through x . Since,by Theorem 3.2, Y (1) is isometrically embedded into X , this implies that for anyvertex x ∈ Min X (h n ) ⊂ Y ⊂ X, there exists an h n -invariant geodesic passingthrough x . (cid:3) Classifying spaces with virtually cyclic stabilisers for -locatedgroups In this section we construct a low-dimensional classifying space for the family ofvirtually cyclic subgroups of a group acting properly on an 8-located complex withthe SD ′ -property. The proof relies on the fact that the minimal displacement set ofsuch complex is a systolic subcomplex that embeds isometrically into the complex.We start by giving the systolic analogue of one of the main results the constructionwill be based on. Theorem 4.1.
Let X be a systolic locally finite simplicial complex. For a hyper-bolic isometry h whose minimal displacement set is a union of axes (that is, for h satisfying (2 . ) and | h | > , the graph of axes ( Y ( h ) , d Y ( h ) ) is quasi-isometric toa simplicial tree. For the proof see [26], Corollary 4 .
6, page 21.
Theorem 4.2.
Let X be a locally finite -located complexes with the SD ′ -property.For a hyperbolic isometry h whose minimal displacement set is a union of axes(that is, for h satisfying (2 . ) and | h | > , the graph of axes ( Y ( h ) , d Y ( h ) ) isquasi-isometric to a simplicial tree.Proof. Let Y = Min X (h). Lemma 3.3 implies that Y = Min Y (h). If there do notexist h -invariant geodesics in X , take an h n -invariant geodesic in X , n > . h -invariant geodesics in X . These geodesics arealso in Y because, according to (2 . Y = span { S γ | γ is an h -invariant geodesic } .Theorems 3.2 and 3.3 imply that Y is systolic and Y (1) embeds isometrically into X . Then, by Theorem 4.1, the result follows. (cid:3) For the rest of the section, let G be a group acting properly discontinuously ona uniformly locally finite 8-located complex X with the SD ′ -property of dimension d . Theorem 4.3.
The systolic complex X is a model for EG . For the proof see [5], Theorem E.In order to construct models for the commensurators N G [ H ], first we show thatthe group G satisfies condition (C). Using this, in every finitely generated subgroup K ⊆ N G [ H ] that contains H we find a suitable normal cyclic subgroup. As shownin [26] for systolic complexes, the quotient group acts properly on a quasi-tree. Lemma 4.4.
The group G satisfies condition (C).Proof. The proof is similar to the one given in [26] (Lemma 5 . g, h ∈ G such that | h | = ∞ , and assume there are k, l ∈ Z such that g − h k g = h l .We have to show that | k | = | l | . Since the action of G on X is proper, the element h acts as a hyperbolic isometry. By Theorem 3.6, there is in X an h n -invariantgeodesic for some n ≥
1. We get the claim by considering the following sequenceof equalities for the translation length: | k | · | h n | = | h k · n | = | g − · h n · k · g | = | h ± l · n | = | l | · | h n | . -LOCATED SIMPLICIAL COMPLEXES WITH THE SD’-PROPERTY 15 The first and the last of the equalities follow from the fact that the translationlength of an element can be measured on an invariant geodesic, the second one isan easy calculation and the third one is straightforward. (cid:3)
Lemma 4.5.
Let K be a finitely generated subgroup of G , and h ∈ K a hyperbolicisometry satisfying (2 . , such that h h i is normal in K . Then the proper action of G on X induces a proper action of K/ h h i on the graph of axes Y ( h ) .Proof. The proof is similar to the one given in [26], Lemma 5 . (cid:3) Lemma 4.6.
Let h be a hyperbolic isometry of an -located complex with the SD ′ -property X . Assume that | h | > . Then if h satisfies (2 . then so does h n for any n ∈ Z \ { } .Proof. The result follows by Lemma 3.10 and the fact that an h -invariant geodesicis h n -invariant. (cid:3) Lemma 4.7.
Let K be a finitely generated subgroup of N G [ H ] that contains H .Then there is a short exact sequence → h h i → K → K/ h h i → , such that h ∈ H is of infinite order and the group K/ h h i is virtually free.Proof. The proof is similar to the one given in [26], Lemma 5 .
4. Choose an elementof infinite order ˜ h ∈ H satisfying the following two conditions:(1) the set Min X (˜h) is the union of axes (see (2 . | ˜ h | > h to a sufficiently large power.Indeed, by Lemma 3.6, there exists n ≥ h n satisfies the first conditionabove. If | ˜ h n | ≤ h n . The element ˜ h n satisfies both con-ditions. If an element satisfies the conditions above then, by Lemma 4.6, so doesany of its powers. Since G satisfies condition (C), by Lemma 2.7, there exists aninteger k ≥ h ˜ h k i is normal in K .Put h = ˜ h k . By Lemma 4.5, the group K/ h h i acts properly by isometries on thegraph of axes ( Y ( h ) , d Y ( h ) ), which, by Theorem 4.2, is a quasi-tree. In conclusionthe group K/ h h i is virtually free. (cid:3) The proofs of the next results are similar to the one given for systolic complexesin [26] (see Lemma 5 .
5, Theorem C).
Lemma 4.8.
For every [ H ] ∈ VCY \ FIN there exist:(1) a -dimensional model for E G [ H ] N G [ H ] ;(2) a -dimensional model for EN G [ H ] . Theorem 4.9.
There exists a model for EG of dimension dim EG = ( d+1 , if d ≤ ,d , if d ≥ . References [1] B. Breˇsar, J. Chalopin, V. Chepoi, T. Gologranc and D. Osajda,
Bucolic complexes
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