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We prove for non-elementary torsion-free hyperbolic groups and all r ≥ 2 that the higher topological complexity TC () is equal to r · cd(). In particular, hyperbolic groups satisfy the rationality conjecture on the TC-generating function, giving an afﬁr- mative answer to a question of Farber and Oprea. More generally, we show that the same conclusions hold for certain toral relatively hyperbolic groups. Keywords Higher topological complexity · Hyperbolic groups · Classifying spaces for families of subgroups Mathematics Subject Classiﬁcation 55M30 · 55R35 · 20F67 1 Introduction Let r ≥ 2 be an integer. The higher (or sequential) topological complexity TC (X ) of a path-connected space X was introduced by Rudyak (2010), generalising Farber’s topological complexity (Farber 2003). The motivation for these numerical invari- ants arises from robotics. They provide a measure of complexity for the motion planning problem in the conﬁguration space X with prescribed initial and ﬁnal states, as well as r − 2 consecutive intermediate states. More precisely, consider [0,1] r the path-ﬁbration p : X → X that maps a path ω:[0, 1]→ X to the tuple 1 r −2 (ω(0), ω( ),...,ω( ), ω(1)). Then TC (X ) is deﬁned as the minimal integer n r −1 r −1 for which X can be covered by n + 1 many open subsets U ,..., U such that p 0 n Kevin Li kevin.li@soton.ac.uk Sam Hughes sam.hughes@maths.ox.ac.uk Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford OX2 6GG, UK School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK 123 324 S. Hughes, K. Li admits a local section over each U . If no such n exists, one sets TC (X ) :=∞.Note i r that TC (X ) recovers the usual topological complexity. Since the higher topological complexities are homotopy invariants, one obtains interesting invariants of groups by setting TC () := TC (K (, 1)), where K (, 1) r r is an Eilenberg–MacLane space. The topological complexities TC () have been com- puted for several classes of groups (see e.g. Farber et al. 2019; Farber and Mescher 2020; Dranishnikov 2020 for r = 2, Farber and Oprea 2019; Aguilar-Guzmán et al. 2021; González et al. 2016 for r ≥ 2, and references therein). In a celebrated result of Dranishnikov (2020) (see also Farber and Mescher 2020), the topological complexity TC () of groups with cyclic centralisers, such as hyperbolic groups, was shown to equal cd( × ). Here cd denotes the cohomological dimension. We generalise this result to all higher topological complexities TC () for r ≥ 2, as well as to a larger class of groups containing certain toral relatively hyperbolic groups. Recall that a col- lection {P | i ∈ I } of subgroups of is called malnormal, if for all g ∈ and i , j ∈ I −1 we have gP g ∩ P ={e}, unless i = j and g ∈ P . i j i Theorem 1.1 Let r ≥ 2 and let be a torsion-free group with cd() ≥ 2. Suppose that admits a malnormal collection of abelian subgroups {P | i ∈ I } satisfying r r cd(P )< cd( ) such that the centraliser C (g) is cyclic for every g ∈ that is not conjugate into any of the P . Then TC () = cd( ). i r The preceding theorem was obtained for the case r = 2 by the second author in Li (2021). For a space X,the TC-generating function f (t ) is deﬁned as the formal power series f (t ) := TC (X ) · t . X r +1 r =1 The TC-generating function of a group is set to be f (t ) := f (t ). Recall that K (,1) a group is said to be of type F (or geometrically ﬁnite) if it admits a ﬁnite model for K (, 1). Following Farber and Oprea (2019), we say that a ﬁnite CW-complex X (resp. a group of type F) satisﬁes the rationality conjecture if the TC-generating P(t ) function f (t ) (resp. f (t )) is a rational function of the form , where P(t ) is an X 2 (1−t ) integer polynomial with P(1) = cat(X ) (resp. P(1) = cd()). Here cat denotes the Lusternik–Schnirelmann category. While a counter-example to the rationality con- jecture for ﬁnite CW-complexes was found in Farber et al. (2020), the rationality conjecture for groups of type F remains open. It is known to hold, e.g. for abelian groups of type F, right-angled Artin groups, fundamental groups of closed orientable surfaces, and Higman’s group (see Farber and Oprea 2019, Section 8). Our result extends the class of groups for which the rationality conjecture holds as follows. Corollary 1.2 Let be a group as in Theorem 1.1.If is of type F, then (2 − t )t f (t ) = cd() . (1 − t ) 123 Higher topological complexity of hyperbolic groups 325 In particular, the rationality conjecture holds for . As remarked by Farber and Oprea in (2019, p. 159), it is particularly interesting to determine the validity of the rationality conjecture for the class of hyperbolic groups. We answer their question in the afﬁrmative. Corollary 1.3 The rationality conjecture holds for torsion-free hyperbolic groups. 2 Background We recall the notion of a classifying space for a family of subgroups (see e.g. Lück 2005). A family F of subgroups of a group G is a non-empty set of subgroups that is closed under conjugation and ﬁnite intersections. The family consisting only of the trivial subgroup is denoted by TR. The family F H generated by a set of subgroups H is the smallest family containing H. For a family F of subgroups of G and a subgroup H of G, we denote by F | the family {L ∩ H | L ∈ F } of subgroups of H.A classifying space E G for the family F is a terminal object in the G-homotopy category of G-CW-complexes with isotropy groups in F. Note that a model for E G TR is given by EG. In particular, for every family F containing the trivial subgroup there exists a G-map EG → E G that is unique up to G-homotopy. Let E ⊂ F be two families of subgroups of G. We say that G satisﬁes condi- tion (M ) if every element in F \E is contained in a unique maximal element E ⊂F M in F \E, and that G satisﬁes condition (NM ) if additionally M equals its nor- E ⊂F maliser N (M ). We denote by W (M ) := N (M )/M the Weyl group of M.The G G G following proposition is a special case of a construction due to (Lück and Weiermann 2012, Corollary 2.8) stated in Li (2021, Corollary 2.2). Theorem 2.1 (Lück–Weiermann) Let G be a group and E ⊂ F be two families of subgroups. Let {M | i ∈ I } be a complete set of representatives for the conjugacy classes of maximal elements in F \E. (i) If E = TR and G satisﬁes condition (M ), then a model for E G is given TR⊂F F by the following G-pushout G × E (N (M )) EG N (M ) G i i ∈I G i G × E (W (M )) E G; N (M ) G i i ∈I G i (ii) If G satisﬁes conditions (M ) and (NM ), then a model for E G is given E ⊂F E ⊂F F by the following G-pushout G × E M E G M i E | E i ∈I i M G/M E G. i F i ∈I 123 326 S. Hughes, K. Li The F-restricted orbit category O G has G-sets G/H with H ∈ F as objects and G-maps as morphisms. Let A be an O G-module, that is a contravariant functor from the orbit category O G to the category of modules. The equivariant cellular cohomology H (X ; A) of a G-CW-complex X with isotropy groups in F is called Bredon cohomology (Bredon 1967). In particular, Bredon cohomology satisﬁes the Mayer–Vietoris axiom for G-pushouts. Moreover, if X is of the form G × Y for a subgroup H ⊂ G and an H -CW-complex Y , then there is a natural induction isomorphism (see e.g. Lück 2005) ∗ ∗ H (G × Y ; A) = H (Y ; A). (1) G H Here on the right hand side, A is regarded as an O H -module by restriction along F | the obvious functor O H → O G. F | F The (higher) topological complexity TC () of a group for r ≥ 2 can be charac- terised in terms of classifying spaces for families (Farber and Oprea 2019, Theorem 3.1), generalising a result of Farber et al. (2019, Theorem 3.3) for the case r = 2. r r Consider G = and let D be the family of subgroups of that is generated by the diagonal subgroup () and the trivial subgroup. Theorem 2.2 (Farber–Oprea) Let be a group and r ≥ 2. Then TC () equals the inﬁmum of integers n for which the canonical -map r r E ( ) → E ( ) r r r is -equivariantly homotopic to a -map with values in the n-skeleton of E ( ). As a consequence (Farber and Oprea 2019, Theorem 5.1), a lower bound for TC () is given by the supremum of integers n for which the canonical -map r r E ( ) → E ( ) induces a non-trivial map in Bredon cohomology n r n r H (E ( ); A) → H (E ( ); A) r r for some O ( )-module A. 3 Proofs Fix an integer r ≥ 2. Let be a group and : → be the diagonal map. For r −1 r γ = (γ ,...,γ ) ∈ and a subset S ⊂ , we deﬁne the subgroup H of 1 r −1 γ,S as −1 −1 H := (γ ,...,γ , e) · (C (S)) · (γ ,...,γ , e). γ,S 1 r −1 1 r −1 Here C (S) denotes the centraliser of S in .For b ∈ , we write H instead of γ,b r −1 H . Denote the element e := (e,..., e) ∈ and note that H = ().The γ,{b} e,e elementary proof of the following lemma is omitted. r −1 Lemma 3.1 Let γ = (γ ,...,γ ) ∈ and S ⊂ be a subset. The following 1 r −1 hold: 123 Higher topological complexity of hyperbolic groups 327 r −1 −1 r −1 (i) For g = (g ,..., g ) ∈ ,let γ := (g γ g ,..., g γ g ) ∈ and 1 r 1 1 r −1 r −1 r r −1 S :={g sg ∈ | s ∈ S}⊂ . Then we have −1 g · H · g = H ; γ,S γ ,S r −1 (ii) For δ = (δ ,...,δ ) ∈ and a subset T ⊂ , we have 1 r −1 H ∩ H = H −1 −1 ; γ,S δ,T γ,S∪T ∪{δ γ ,...,δ γ } 1 r −1 1 r −1 (iii) We have −1 −1 r N (H ) ={(γ k hγ ,...,γ k hγ , h) ∈ | h ∈ N (C (S)), γ,S 1 1 r −1 r −1 1 r −1 k ,..., k ∈ C (C (S))}. 1 r −1 Let F ⊂ D be the families of subgroups of deﬁned as D := F {()} ∪ TR; r −1 F = F {H | γ ∈ , b ∈ \{e}} ∪ TR. 1 γ,b From now on, let be a group as in Theorem 1.1. The following properties of centralisers in will be used in the sequel. For b, c ∈ \{e}, we have either C (b) = C (c) or C (b) ∩ C (c) ={e} by (Li 2021, Lemma 3.4). By assumption on , the centraliser C (b) of an element b ∈ \{e} is inﬁnite cyclic or isomorphic to one of the P . In both cases we have N (C (b)) = C (C (b)) = C (b) (see Li 2021, Lemma 3.5) for more details. r −1 Lemma 3.2 Let be a group as in Theorem 1.1 and let e = (e,..., e) ∈ . Then r r for n = cd( ) and every O ( )-module A, we have n r H × E (H ); A = 0. H F | e,e e,e 1 H e,e Proof We have that conditions (M ) and (NM ) hold for the TR⊂F | TR⊂F | 1 H 1 H e,e e,e group H . To see this, we note that under the obvious isomorphism H ,the e,e e,e family F | is identiﬁed with F {C (b) | b ∈ \{e}} ∪ TR, using Lemma 3.1(i) 1 H e,e and (ii). By Theorem 2.1(ii), we obtain an H -pushout e,e H × E (H ) E (H ) e,e H e,b e,e H ∈M e,b e,b (2) H /H E (H ), e,e e,b F | e,e H ∈M 1 H e,b e,e where M is a complete set of representatives of conjugacy classes of maximal elements in F | \TR. Since cd(H )< n and cd(H )< n − 1for b ∈ \{e},the 1 H e,e e,b e,e Mayer–Vietoris sequence for H (−; A) applied to the pushout (2) together with the e,e induction isomorphism (1) yields the lemma. 123 328 S. Hughes, K. Li r r Proof of Theorem 1.1 Throughout the proof let n = cd( ). We show that the -map n n r r r r E ( ) → E ( ) induces a surjective map H (E ( ); A) → H (E ( ); A) for r r D D every O ( )-module A. Then the theorem follows from Theorem 2.2 and the upper bound TC () ≤ n (see e.g. Farber and Oprea 2019, (4)). First, observe that condition (M ) holds, since conjugates of the subgroups TR⊂F generating F either coincide or have trivial intersection by Lemma 3.1(i) and (ii). r −1 r Moreover, for γ ∈ and b ∈ \{e} there is an isomorphism N (H ) = C (b) γ,b by Lemma 3.1(iii). In particular, we have cd(N (H )) < n, since γ,b r −1 n ≥ cd( × Z ) = cd() + r − 1 > r and n > cd(P ) by assumption. Then it follows from the induction isomorphism (1) that we have n r H × E (N r (H )); A = 0 N r (H ) γ,b γ,b r r for every O ( )-module A. By Theorem 2.1(i), we obtain a -pushout r r × E (N (H )) E ( ) N r (H ) γ,b H ∈M γ,b γ,b (3) r r × E (W (H )) E ( ), N r (H ) γ,b F H ∈M γ,b 1 γ,b where M is a complete set of representatives of conjugacy classes of maximal elements in F \TR. Applying the Mayer–Vietoris sequence for H (−; A) to the pushout (3) n r n r shows that the induced map H (E ( ); A) → H (E ( ); A) is surjective. r r Second, observe that condition (M ) holds, since by Lemma 3.1(i) and (ii), F ⊂D two conjugates of () in either coincide or their intersection lies in F . Moreover, condition (NM ) holds by Lemma 3.1(iii), using that the centre C () of is F ⊂D trivial. By Theorem 2.1(ii), we obtain a -pushout r r × E (H ) E ( ) H F | e,e F e,e 1 1 He,e (4) r r /H E ( ). e,e Then Lemma 3.2 and the Mayer–Vietoris sequence for H (−; A) applied to the n r n r pushout (4) yield that the map H (E ( ); A) → H (E ( ); A) is surjective. r r D F n r n r Together, the map H (E ( ); A) → H (E ( ); A) is surjective for every r r O ( )-module A. This ﬁnishes the proof. Proof of Corollary 1.2 For groups of type F,wehavecd( ) = r · cd() by Dran- ishnikov (2019, Corollary 2.5). The result now follows from Theorem 1.1. 123 Higher topological complexity of hyperbolic groups 329 Proof of Corollary 1.3 The result follows from Corollary 1.2 using the fact that torsion- free hyperbolic groups are of type F (see e.g. Bridson and Haeﬂiger 1999, Corollary 3.26 on p. 470). Acknowledgements Both authors would like to thank the organisers of the workshop “Advances in Homo- topy Theory” held in September 2021 at the Southampton Centre for Geometry, Topology, and Applications which inspired this work. The ﬁrst author was supported by the Engineering and Physical Sciences Research Council (Grant Number 2127970) and the European Research Council under the European Union’s Hori- zon 2020 research and innovation programme (Grant Number 850930). We thank the referees for helpful comments. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Aguilar-Guzmán, J., González, J., Oprea, J.: Right-angled Artin groups, polyhedral products and the TC- generating function. Proc. R. 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Journal of Applied and Computational Topology – Springer Journals
Published: Sep 1, 2022
Keywords: Higher topological complexity; Hyperbolic groups; Classifying spaces for families of subgroups; 55M30; 55R35; 20F67
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