Presentations of higher dimensional Thompson groups

Brin, Matthew G.

doi:10.1016/j.jalgebra.2004.10.028

Cited by 15 publications

(69 citation statements)

References 9 publications

(35 reference statements)

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“…These groups can be considered as an n-dimensional analogue of the Higman-Thompson group V k,r = 1V k,r . In [5,6], he proved that V k,r and 2V 2,1 are not isomorphic, 2V 2,1 is simple and 2V 2,1 is finitely presented. He also proved that nV 2,1 is simple for all n ∈ N in [7].…”

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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Section: Classificationmentioning

confidence: 99%

Étale groupoids arising from products of shifts of finite type

Matui

2016

Advances in Mathematics

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“…Brin [2004;2005] defines nV as a subgroup of the homeomorphism group of the n-fold product of the Cantor set with itself, and then shows there are a number of equivalent characterizations. The one we use here is Brin's characterization [2005] in terms of equivalence classes of labeled tree pair diagrams.…”

Section: Background On Higher-dimensional Thompson's Groupsmentioning

confidence: 99%

“…A tree-pair diagram for the identity element that has a nonidentity permutation. Brin [2004;2005] showed that the groups nV are finitely generated and gave several generating sets for 2V. As is common with groups of the Thompson family, there is an infinite presentation, which is useful for its symmetry and regularity and which has a finite subpresentation.…”

Section: Background On Higher-dimensional Thompson's Groupsmentioning

confidence: 99%

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Metric properties of higher-dimensional Thompson’s groups

Burillo¹,

Cleary²

2010

Pacific J. Math.

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Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic selfmaps of n-dimensional unit cubes. We describe some of the metric properties of higherdimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.

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“…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 .…”

mentioning

confidence: 99%

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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Presentations of higher dimensional Thompson groups

Cited by 15 publications

References 9 publications

Étale groupoids arising from products of shifts of finite type

Étale groupoids arising from products of shifts of finite type

Metric properties of higher-dimensional Thompson’s groups

The Brin–Thompson groupssVare of type F_∞

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Presentations of higher dimensional Thompson groups

Cited by 15 publications

References 9 publications

Étale groupoids arising from products of shifts of finite type

Étale groupoids arising from products of shifts of finite type

Metric properties of higher-dimensional Thompson’s groups

The Brin–Thompson groupssVare of type F∞

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Product

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The Brin–Thompson groupssVare of type F_∞