Cohomological finiteness properties of the Brin–Thompson–Higman groups 2<i>V</i> and 3<i>V</i>

Kochloukova, Dessislava H.; Martı́nez-Pérez, Conchita; Nucinkis, Brita E. A.

doi:10.1017/s0013091513000369

Cited by 18 publications

(50 citation statements)

References 8 publications

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“…This is exactly the condition that proved difficult to verify for the filtration of |P 1 | in [KMPN10].…”

Section: The Stein Space For Svmentioning

confidence: 92%

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The Brin–Thompson groupssVare of type F_∞

Fluch¹,

Marschler²,

Witzel³

et al. 2013

Pacific J. Math.

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Abstract. We prove that the Brin-Thompson groups sV , also called higher dimensional Thompson's groups, are of type F∞ for all s ∈ N. This result was previously shown for s ≤ 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is easier to analyze.Recall that a group is of type F ∞ if it admits a classifying space with finitely many cells in each dimension. Well-known examples of groups of type F ∞ include Thompson's groups F , T , and V . Some generalizations of V were introduced by Brin [Bri04, Bri05] and shown to be simple. We denote these groups sV , for s ∈ N, with 1V = V . These groups are usually termed higher-dimensional Thompson's groups or Brin-Thompson groups. All of the groups sV are known to be finitely presented [HM12], and Kochloukova, Martínez-Pérez, and Nucinkis [KMPN10] showed that 2V and 3V are of type F ∞ . We prove that this result extends to all dimensions.Main Theorem. The Brin-Thompson group sV is of type F ∞ for all s.Fix some s. There is a natural poset P 1 associated to sV . The realization |P 1 | of this poset is contractible and the action of sV is proper but not cocompact. To prove the Main Theorem it suffices to produce a cocompact filtration of |P 1 | whose connectivity tends to infinity. The tool to study relative connectivity is discrete Morse theory. This was carried out for s = 2, 3 in [KMPN10]. However, for larger s this space quickly becomes cumbersome.We therefore consider a subspace sX of |P 1 | which we call the Stein space for sV . As before, the Stein space is contractible and the action is not cocompact. The advantage of the Stein space is that the Morse theory becomes much easier to handle.The paper is organized as follows. In Section 1 we recall the definition of sV . The Stein space sX is defined in Section 2 and some basic properties are verified. In Section 3 we analyze the connectivity of the subspaces in the filtration and deduce the Main Theorem.Acknowledgments. We would like to thank Kai-Uwe Bux for suggesting this project and for many helpful discussions. We also gratefully acknowledge support through the SFB 701 in Bielefeld (first, second, and fourth author) and the SFB 878 in Münster (third author). The research for this article was carried out during the 2012 PCMI Summer Session in Geometric Group Theory and we thank the organizers and the PCMI for this opportunity. Finally, we thank Matt Brin for his helpful suggestions and remarks.

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“…This is exactly the condition that proved difficult to verify for the filtration of |P 1 | in [KMPN10].…”

Section: The Stein Space For Svmentioning

confidence: 92%

“…These groups are usually termed higher-dimensional Thompson's groups or Brin-Thompson groups. All of the groups sV are known to be finitely presented [HM12], and Kochloukova, Martínez-Pérez, and Nucinkis [KMPN10] showed that 2V and 3V are of type F ∞ . We prove that this result extends to all dimensions.…”

mentioning

confidence: 99%

The Brin–Thompson groupssVare of type F_∞

Fluch¹,

Marschler²,

Witzel³

et al. 2013

Pacific J. Math.

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“…By using his method, we proved that the topological full group [[G A ]] of any SFT groupoid G A is of type F ∞ ([28, Theorem 6.21]). As for the higher dimensional Thompson groups, D. H. Kochloukova, C. Martínez-Pérez and B. E. A. Nucinkis [22] showed that 2V 2,1 and 3V 2,1 are of type F ∞ . This was later generalized to all nV 2,1 by M. Fluch, M. Marschler, S. Witzel and M. C. B. Zaremsky [12].…”

Section: Classificationmentioning

confidence: 99%

Étale groupoids arising from products of shifts of finite type

Matui

2016

Advances in Mathematics

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Two conjectures about homology groups, K-groups and topological full groups of minimalétale groupoids on Cantor sets are formulated. We verify these conjectures for many examples ofétale groupoids including products ofétale groupoids arising from one-sided shifts of finite type. Furthermore, we completely determine when these product groupoids are mutually isomorphic. Also, the abelianization of their topological full groups are computed. They are viewed as generalizations of the higher dimensional Thompson groups.

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“…The three Thompson groups F , T , and V have type F ∞ [4,7], as do many of their variants such as the generalized groups F n,k , T n,k and V n,k [4], certain diagram groups [12] and picture groups [14], braided Thompson groups [8], higher-dimensional groups nV [16,20], and various other generalizations [1,10,11,15,21,22].…”

Section: Introductionmentioning

confidence: 99%

Röver's simple group is of type $F_\infty$

Belk¹,

Matucci²

2016

Publicacions Matemàtiques

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Abstract. We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimiplical complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

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Cohomological finiteness properties of the Brin–Thompson–Higman groups 2V and 3V

Abstract: Abstract. We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

Cited by 18 publications

References 8 publications

The Brin–Thompson groupssVare of type F_∞

The Brin–Thompson groupssVare of type F_∞

Étale groupoids arising from products of shifts of finite type

Röver's simple group is of type $F_\infty$

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Cohomological finiteness properties of the Brin–Thompson–Higman groups 2V and 3V

Abstract: Abstract. We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

Cited by 18 publications

References 8 publications

The Brin–Thompson groupssVare of type F∞

The Brin–Thompson groupssVare of type F∞

Étale groupoids arising from products of shifts of finite type

Röver's simple group is of type $F_\infty$

Contact Info

Product

Resources

About

The Brin–Thompson groupssVare of type F_∞

The Brin–Thompson groupssVare of type F_∞