Presentations for the higher-dimensional Thompson groups<i>nV</i>

Hennig, Johanna; Matucci, Francesco

doi:10.2140/pjm.2012.257.53

Cited by 25 publications

(46 citation statements)

References 7 publications

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“…Proof. We can prove this proposition exactly in the same way as [6] and [19], using k(d)-ary trees instead of binary trees.…”

Section: Clearly One Hasmentioning

confidence: 84%

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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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“…Proof. We can prove this proposition exactly in the same way as [6] and [19], using k(d)-ary trees instead of binary trees.…”

Section: Clearly One Hasmentioning

confidence: 84%

“…As mentioned in Remark 5.13, the topological full group of G [2] × G [2] is the two dimensional Thompson group 2V 2,1 . Presentations of the group 2V 2,1 was given by M. G. Brin [6], and later it was extended to nV 2,1 by J. Hennig and F. Matucci [19]. We have to generalize some arguments of these works to…”

Section: Abelianizationmentioning

confidence: 99%

Étale groupoids arising from products of shifts of finite type

Matui

2016

Advances in Mathematics

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“…They were first defined by Matt Brin in [7], and can be viewed as higher-dimensional generalizations of the group V (= 1V ) defined by Richard J. Thompson (see [10] for an introduction to Thompson's groups). The groups nV are finitely presented [19] and simple [9], and for 1 ≤ j < k it is known that jV embeds into kV [7], but also that jV and kV are not isomorphic [3].…”

Section: Introductionmentioning

confidence: 99%

Embedding right-angled Artin groups into Brin–Thompson groups

Belk¹,

Bleak²,

Matucci³

2019

Math. Proc. Camb. Phil. Soc.

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We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group nV . It follows that many other groups can be embedded into some nV (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of nV are unsolvable.The third author gratefully acknowledges the Fundação para a Ciência e a Tecnologia (FCT -PEst-OE/MAT/UI0143/2014) for the support received during the development of this work.

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“…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.…”

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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 for the higher-dimensional Thompson groupsnV

Cited by 25 publications

References 7 publications

Étale groupoids arising from products of shifts of finite type

Étale groupoids arising from products of shifts of finite type

Embedding right-angled Artin groups into Brin–Thompson groups

The Brin–Thompson groupssVare of type F_∞

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Presentations for the higher-dimensional Thompson groupsnV

Cited by 25 publications

References 7 publications

Étale groupoids arising from products of shifts of finite type

Étale groupoids arising from products of shifts of finite type

Embedding right-angled Artin groups into Brin–Thompson groups

The Brin–Thompson groupssVare of type F∞

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Product

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The Brin–Thompson groupssVare of type F_∞