On the topological complexity of toral relatively hyperbolic groups
aa r X i v : . [ m a t h . A T ] F e b ON THE TOPOLOGICAL COMPLEXITY OF TORALRELATIVELY HYPERBOLIC GROUPS
KEVIN LI
Abstract.
We prove that the topological complexity TC( π ) equals cd( π × π )for certain toral relatively hyperbolic groups π . Introduction
The (reduced) topological complexity TC( X ) of a space X is defined as theminimal integer n for which there exists a cover of X × X by n + 1 open subsets U , . . . , U n such that the path fibration X [0 , → X × X admits a local sectionover each U i . This quantity, which is similar in spirit to the classical Lusternik–Schnirelmann category, was introduced by Farber [Far03] in the context of robotmotion planning. In fact, TC( − ) is a homotopy invariant and hence one can definethe topological complexity TC( π ) of a group π to be TC( Bπ ), where Bπ is theclassifying space for π . There are bounds cd( π ) ≤ TC( π ) ≤ cd( π × π ), wherecd( − ) denotes the cohomological dimension. However, the precise value of TC( π )is known only for a small class of groups, which contains for instance the abeliangroups, hyperbolic groups, free products of the form H ∗ H for H geometricallyfinite, right-angled Artin groups, and certain subgroups of braid groups. We referto [FM20] and [Dra20] for a more thorough account on this topic.It is the decisive insight of [FGLO19] that the topological complexity of groupscan be expressed in terms of classifying spaces for families of subgroups, whichare well-studied objects in equivariant topology. For a family F of subgroups ofa group G , the classifying space E F G is a terminal object, up to G -homotopy,among G -CW-complexes with stabilizers in F . Farber, Grant, Lupton, and Opreashowed that TC( π ) equals the minimal integer n for which the canonical ( π × π )-map E ( π × π ) → E D ( π × π ) is equivariantly homotopic to a map with values in the n -skeleton E D ( π × π ) ( n ) . Here D is the family of subgroups of π × π consisting of allconjugates of the diagonal subgroup ∆( π ) and their subgroups. Using this charac-terization of TC( π ), in a recent breakthrough Dranishnikov [Dra20] has computedthe topological complexity of torsionfree hyperbolic groups and more generally, ofgeometrically finite groups with cyclic centralizers. Theorem 1.1 (Dranishnikov) . Let π be a geometrically finite group with cd( π ) ≥ such that the centralizer Z π ( b ) is cyclic for any b ∈ π \{ e } . Then TC( π ) = cd( π × π ) . Recall that a group π is called geometrically finite if it admits a finite modelfor Bπ . Note that for geometrically finite groups π we have cd( π × π ) = 2 cd( π ), Date : March 1, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Topological complexity, toral relatively hyperbolic groups. see [Dra19]. Previously, Farber and Mescher [FM20] had shown for groups π asin Theorem 1.1 that TC( π ) equals either cd( π × π ) or cd( π × π ) −
1. The maincontribution of the present note is the following generalization of Theorem 1.1.
Theorem 1.2.
Let π be a torsionfree group with cd( π ) ≥ . Suppose that π admitsa malnormal collection of abelian subgroups P = { P i | i ∈ I } satisfying cd( P i × P i ) < cd( π × π ) such that the centralizer Z π ( b ) is cyclic for any b ∈ π that is not conjugateinto any of the P i . Then TC( π ) = cd( π × π ) . Recall that a set P = { P i | i ∈ I } of subgroups of π is called a malnormalcollection if for any P i , P j ∈ P and g ∈ π , we have gP i g − ∩ P j = { e } or i = j and g ∈ P i . Our main examples of groups satisfying the assumptions of Theorem 1.2are torsionfree relatively hyperbolic groups π with cd( π ) ≥ P , . . . , P k satisfying cd( P i ) < cd( π ). Note that The-orem 1.2 recovers Theorem 1.1 as a special case when P consists only of the trivialsubgroup and that the assumption of geometric finiteness has been dropped.In light of the upper bound TC( π ) ≤ cd( π × π ), Theorem 1.1 and Theorem 1.2are statements about the maximality of topological complexity. They share a com-mon strategy of proof based on the characterization of TC( π ) in terms of classify-ing spaces from [FGLO19]. Namely, we construct a “small” model for E D ( π × π )from E ( π × π ) allowing us to show that the map E ( π × π ) → E D ( π × π ) inducesa non-trivial map on cohomology in degree cd( π × π ). Hence one has equalityTC( π ) = cd( π × π ). Nevertheless, even for the case when P consists only of thetrivial subgroup, our proof is different from Dranishnikov’s. He constructed a spe-cific model for E D ( π × π ) and used cohomology with compact support, while weemploy a general construction due to L¨uck and Weiermann and use equivariant Bre-don cohomology. L¨uck and Weiermann’s construction (Theorem 2.1) is a generalrecipe to efficiently construct E F G from E E G for two families of subgroups E ⊂ F of a group G satisfying a certain maximality condition. While for the group π × π this condition is not satisfied for the families ⊂ D , we define an intermediatefamily ⊂ F ⊂ D such that we can apply two iterations of the construction. Acknowledgments.
The present note is part of the author’s PhD project underthe supervision of Nansen Petrosyan and Ian Leary, who we thank for their sup-port. We are grateful to Pietro Capovilla for interesting discussions about thepaper [CLM] and his master’s thesis. We thank Mark Grant for helpful commentson an earlier version of this note.2.
Preliminaries on classifying spaces for families
We briefly review the notion of classifying spaces for families of subgroups dueto tom Dieck and their equivariant Bredon cohomology. For a survey on classifyingspaces for families we refer to [L¨uc05] and for an introduction to Bredon cohomologyto [Flu]. Let G be a group, which shall always mean a discrete group.A family of subgroups F is a non-empty set of subgroups of G that is closedunder conjugation by elements of G and under taking subgroups. Typical examplesare T R = , FIN = { finite subgroups } , VCY = { virtually cyclic subgroups } ,and ALL = { all subgroups } . For a set H of subgroups of G , one can consider FhHi = { conjugates of subgroups in H and their subgroups } which is the smallestfamily containing H and called the family generated by H . When H = { H } consists OPOLOGICAL COMPLEXITY OF TORAL RELATIVELY HYPERBOLIC GROUPS 3 of a single subgroup, we denote
Fh{ H }i instead by Fh H i and call it the familygenerated by H . For a family F of subgroups of G and any subgroup H ⊂ G , wedenote by F| H the family { K ∩ H | K ∈ F} of subgroups of H . (In the literaturethis family is sometimes denoted by F ∩ H instead.)A classifying space E F G for the family F is a terminal object in the G -homotopycategory of G -CW-complexes with stabilizers in F . It can be shown that E F G always exists and that a G -CW-complex X is a model for E F G if and only if thefixed-point set X H is contractible for H ∈ F and empty otherwise. In particular,there exists a G -map EG → E F G which is unique up to G -homotopy.The orbit category O F G has as objects G/H for H ∈ F and as morphisms G -maps. Let O F G - Mod denote the category of contravariant functors M : O F G → Z - Mod with values in the category of Z -modules, which are called O F G -modules .For a G -CW-complex X with stabilizers in F , the G -equivariant Bredon cohomol-ogy H ∗ G ( X ; M ) with coefficients in an O F G -module M is the cohomology of thecochain complex Hom O F G - Mod ( C ∗ ( X ? ) , M ), where C ∗ ( X ? )( G/H ) = C ∗ ( X H ) isthe cellular chain complex. Passage to larger families.
Let G be a group and E ⊂ F be two families ofsubgroups.We say that G satisfies condition ( M E⊂F ) if every element H ∈ F \E is containedin a unique element M ∈ F \E which is maximal in F \E (with respect to inclusion).We say that G satisfies condition ( N M
E⊂F ) if every maximal element M ∈ F \ E is self-normalizing, i.e. M equals its normalizer N G M in G . Let M = { M i | i ∈ I } be a complete set of representatives for the conjugacy classes of maximal elementsin F \ E , i.e. each M i is maximal in F \ E and any maximal element in
F \ E is conjugate to precisely one of the M i . The following [LW12, Corollary 2.8] is aspecial case of a more general construction due to L¨uck and Weiermann. Theorem 2.1 (L¨uck–Weiermann) . Let G be a group satisfying condition ( M E⊂F ) for two families of subgroups E ⊂ F . Consider a cellular G -pushout of the form ` i ∈ I G × N G M i E E| NGMi ( N G M i ) E E G ` i ∈ I G × N G M i E ALL| Mi ∪E| NGMi ( N G M i ) X ϕ ` i ∈ I id G × NGMi f i such that each f i is a cellular N G M i -map and ϕ is an inclusion of G -CW-complexes,or such that each f i is an inclusion of N G M i -CW-complexes and ϕ is a cellular G -map. Then X is a model for E F G . Note that a G -pushout as in Theorem 2.1 with maps f i and ϕ as required alwaysexists by using equivariant cellular approximation and mapping cylinders. Corollary 2.2.
Let G be a group and E ⊂ F be two families of subgroups. (i) If G satisfies condition ( M T R⊂F ) , then a model for E F G can be constructedas a G -pushout of the form ` i ∈ I G × N G M i E ( N G M i ) EG ` i ∈ I G × N G M i E ( N G M i /M i ) E F G ; KEVIN LI (ii) If G satisfies conditions ( M E⊂F ) and ( N M
E⊂F ) , then a model for E F G can be constructed as a G -pushout of the form ` i ∈ I G × M i E E| Mi M i E E G ` i ∈ I G/M i E F G .
Proof.
This follows from Theorem 2.1 by observing that if E| N G M i ⊂ ALL| M i ,then a model for E ALL| Mi ∪E| NGMi ( N G M i ) is given by E ( N G M i /M i ) regarded as a N G M i -CW-complex. (cid:3) Homotopy dimension and cohomological dimension of maps.
Let G be agroup and E ⊂ F be two families of subgroups. The following notation is notstandard.We denote by hdim
E⊂F ( G ) the minimal integer n for which the canonical G -map E E G → E F G is G -homotopic to a G -map with values in the n -skeleton ( E F G ) ( n ) .We denote by cd E⊂F ( G ) the maximal integer k for which the induced map onBredon cohomology H kG ( E F G ; M ) → H kG ( E E G ; M ) is non-trivial for some O F G -module M . One clearly has the inequalitycd E⊂F ( G ) ≤ hdim E⊂F ( G ) . Topological complexity as homotopy dimension.
Let π be a group and ∆( π ) ⊂ π × π be the diagonal subgroup. Consider the family D := Fh ∆( π ) i of subgroupsof π × π that is generated by ∆( π ). The following is the main result of [FGLO19,Theorem 3.3]. Theorem 2.3 (Farber–Grant–Lupton–Oprea) . Let π be a group. Then TC( π ) =hdim T R⊂D ( π × π ) . Theorem 2.3 was recently generalized to families generated by a single subgroupin [BCE, Theorem 1.1] and to arbitrary families in [CLM, Proposition 7.5].3.
Structure of the diagonal family of π × π Let π be a group and ∆ : π → π × π be the diagonal map. For a subset S ⊂ π ,denote by Z π ( S ) the centralizer of S in π . The following notation is adoptedfrom [FGLO19] and [Dra20].For γ ∈ π and a subset S ⊂ π , define the subgroup H γ,S of π × π to be H γ,S := ( γ, e ) · ∆( Z π ( S )) · ( γ − , e ) . When S is a singleton set { b } , we write H γ,b instead of H γ, { b } . Note that H e,e =∆( π ). The proof of the following identities is elementary and left to the reader. Lemma 3.1.
Let γ, δ ∈ π and S, T ⊂ π be subsets. Then the following hold: (i) ( g, h ) · H γ,S · ( g − , h − ) = H gγh − ,hSh − for any ( g, h ) ∈ π × π ; (ii) H γ,S ∩ H δ,T = H γ,S ∪ T ∪{ δ − γ } ; (iii) N π × π H γ,S = { ( γkhγ − , h ) ∈ π × π | h ∈ N π ( Z π ( S )) , k ∈ Z π ( Z π ( S )) } . We define the families F ⊂ D of subgroups of π × π to be(1) D := Fh ∆( π ) i ; F := Fh{ H γ,b | γ ∈ π, b ∈ π \ { e }}i . OPOLOGICAL COMPLEXITY OF TORAL RELATIVELY HYPERBOLIC GROUPS 5
In view of Lemma 3.1 (i) and (ii), the family F is generated by the intersectionsof conjugates of the diagonal subgroup ∆( π ). Lemma 3.2.
Let π be a group. Then condition ( M F ⊂D ) holds for the group π × π .Moreover, if the center Z π ( π ) of π is trivial, then condition ( N M F ⊂D ) holds.Proof. If F equals D , then the statement is vacuous, so we may assume that F isstrictly contained in D . For γ ∈ π , conjugates of H γ,e are of the form H δ,e for some δ ∈ π by Lemma 3.1 (i). If γ = δ , then H γ,e ∩ H δ,e ∈ F by Lemma 3.1 (ii). Hencethe { H γ,e | γ ∈ π } are precisely the maximal elements in D \ F and condition( M F ⊂D ) holds. Moreover, given that Z π ( π ) is trivial, we have N π × π ( H γ,e ) = H γ,e by Lemma 3.1 (iii). (cid:3) From now on and for the remainder of this note, we specialize to the followingsituation.
Setup 3.3.
Let π be a torsionfree group admitting a malnormal collection of abeliansubgroups P = { P i | i ∈ I } such that the centralizer Z π ( b ) is cyclic for any b ∈ π that is not conjugate into any of the P i .Note that in the situation of Setup 3.3, we have N π ( Z π ( P i )) = Z π ( P i ) = P i for every P i ∈ P . Our main examples of groups as in Setup 3.3 are torsionfreerelatively hyperbolic groups with finitely generated abelian peripheral subgroups,so-called toral relatively hyperbolic groups.The following lemma for the case when P = can be found in [FGLO19,Lemma 8.0.4] from where the first part of the proof is recalled. Lemma 3.4.
Let π be a group as in Setup 3.3. Then for b, c ∈ π \ { e } , we haveeither Z π ( b ) = Z π ( c ) or Z π ( b ) ∩ Z π ( c ) = { e } .Proof. Let b, c ∈ π \ { e } be two elements. Suppose neither b nor c are conjugateinto any of the P i and that Z π ( b ) ∩ Z π ( c ) is non-trivial. Let Z π ( b ), Z π ( c ) and Z π ( b ) ∩ Z π ( c ) be generated by x , y and z , respectively. Then x n = z = y m for some n, m ∈ Z . Observe that z is not conjugate into any of the P i . Thus its centralizer Z π ( z ) is infinite cyclic and contains both x and y . Therefore, x and y commuteand it follows that Z π ( b ) = Z π ( c ).Suppose b ∈ π \ { e } and c ∈ gP i g − for some g ∈ π , P i ∈ P . Note that Z π ( c ) = gP i g − . If Z π ( b ) ∩ gP i g − is non-trivial, then b ∈ gP i g − by malnormalityof P and hence Z π ( b ) = Z π ( c ). (cid:3) Lemma 3.5.
Let π be a group as in Setup 3.3. Then we have the following: (i) Condition ( M T R⊂F ) holds for the group π × π . Moreover, for γ ∈ π and b ∈ π \ { e } there is an isomorphism N π × π H γ,b ∼ = Z π ( b ) × Z π ( b ) ; (ii) Conditions ( M T R⊂F | He,e ) and ( N M
T R⊂F | He,e ) hold for the group H e,e .Proof. (i) For γ ∈ π and b ∈ π \{ e } , conjugates of H γ,b are of the form H δ,c for some δ ∈ π , c ∈ π \ { e } by Lemma 3.1 (i). We have either H γ,b = H δ,c or H γ,b ∩ H δ,c = { ( e, e ) } by Lemma 3.1 (ii) and Lemma 3.4. Hence the { H γ,b | γ ∈ π, b ∈ π \ { e }} are precisely the maximal elements in F \ T R and condition ( M T R⊂F ) holds.Moreover, for b ∈ π that is not conjugate into any of the P i , observe that N π ( Z π ( b ))is torsionfree virtually cyclic and hence infinite cyclic. It follows that N π ( Z π ( b )) = KEVIN LI Z π ( b ) ∼ = Z . If b ∈ gP i g − for some g ∈ π and P i ∈ P , we have N π ( Z π ( b )) = gP i g − which is abelian and coincides with Z π ( b ). Thus, for any b ∈ π \ { e } we have N π × π H γ,b = { ( γkhγ − , h ) | h, k ∈ Z π ( b ) } ∼ = Z π ( b ) × Z π ( b )by Lemma 3.1 (iii).(ii) Under the isomorphism H e,e ∼ = π , the family F | H e,e is identified with thefamily Fh{ Z π ( b ) | b ∈ π \ { e }}i . The claim follows as before by Lemma 3.4 and theobservation that Z π ( b ) is self-normalizing for any b ∈ π \ { e } . (cid:3) Maximality of topological complexity
The following is the main technical result of this note and will immediately implyTheorem 1.2 from the introduction.
Theorem 4.1.
Let π be a torsionfree group with cd( π ) ≥ . Suppose that π admitsa malnormal collection of abelian subgroups P = { P i | i ∈ I } satisfying cd( P i × P i ) < cd( π × π ) such that the centralizer Z π ( b ) is cyclic for any b ∈ π that is not conjugateinto any of the P i . Then cd T R⊂D ( π × π ) = cd( π × π ) .Proof. We denote cd( π × π ) by n and may assume that it is finite. Consider thefamilies T R ⊂ F ⊂ D of subgroups of π × π as defined in (1).First, condition ( M T R⊂F ) holds by Lemma 3.5 (i) and hence Corollary 2.2 (i)yields a ( π × π )-pushout(2) ` H γ,b ∈M ( π × π ) × N π × π H γ,b E ( N π × π H γ,b ) E ( π × π ) ` H γ,b ∈M ( π × π ) × N π × π H γ,b E ( N π × π H γ,b /H γ,b ) E F ( π × π ) , where M is a complete set of representatives of conjugacy classes of maximalelements in F \ T R . Moreover, in Lemma 3.5 (i) we identified N π × π H γ,b ∼ = Z π ( b ) × Z π ( b ) which is isomorphic to Z × Z or P i × P i for some P i ∈ P and hencehas cohomological dimension strictly less than n . Thus, for any O D ( π × π )-module M , we have H nπ × π (( π × π ) × N π × π H γ,b E ( N π × π H γ,b ); M ) = 0 . Applying the Mayer–Vietoris sequence for H ∗ π × π ( − ; M ) to the pushout (2) yieldsthat the map H nπ × π ( E F ( π × π ); M ) → H nπ × π ( E ( π × π ); M )is surjective.Second, conditions ( M F ⊂D ) and ( N M F ⊂D ) hold by Lemma 3.2 and henceCorollary 2.2 (ii) yields a ( π × π )-pushout(3) ( π × π ) × H e,e E F | He,e ( H e,e ) E F ( π × π )( π × π ) /H e,e E D ( π × π ) . Applying the Mayer–Vietoris sequence for H ∗ π × π ( − ; M ) to the pushout (3) yieldsthat the map H nπ × π ( E D ( π × π ); M ) → H nπ × π ( E F ( π × π ); M ) OPOLOGICAL COMPLEXITY OF TORAL RELATIVELY HYPERBOLIC GROUPS 7 is surjective provided that(4) H nπ × π (( π × π ) × H e,e E F | He,e ( H e,e ); M ) = 0 . The latter is true by another application of Corollary 2.2 (ii) using that conditions( M T R⊂F | He,e ) and (
N M
T R⊂F | He,e ) hold for the group H e,e by Lemma 3.5 (ii). Ityields an H e,e -pushout(5) ` H e,b ∈M ′ H e,e × H e,b E ( H e,b ) E ( H e,e ) ` H e,b ∈M ′ H e,e /H e,b E F | He,e ( H e,e ) , where M ′ is a complete set of representatives of conjugacy classes of maximalelements in F | H e,e \ T R . The Mayer–Vietoris sequence for H ∗ H e,e ( − ; M ) appliedto the pushout (5) shows that (4) indeed holds, using that cd( H e,e ) < n andcd( H e,b ) < n − b ∈ π \ { e } .Together, the map H nπ × π ( E D ( π × π ); M ) → H nπ × π ( E ( π × π ); M )is surjective for any O D ( π × π )-module M . Finally, the coefficients M can be chosensuch that H nπ × π ( E ( π × π ); M ) is non-trivial. This concludes the proof. (cid:3) Proof of Theorem 1.2.
It follows from Theorem 4.1 that the inequalitiescd
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