The Brin–Thompson groupssVare of type F_∞

Abstract

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.

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

“…This proof is the same as the proof in Lemma 2.4 of [16], which itself derives from the proof of the lemma in Section 4 of [5].…”

“…The complex X Stein is the analog of the complexes for F , T , and V introduced by Stein in [28]. Similar complexes were introduced in [8] and [16] for the braided Thompson groups BV and the higherdimensional Thompson groups sV , respectively.…”

“…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].…”

“…Our basic approach is quite similar to that used in [8] for the braided Thompson groups and in [16] for the higher-dimensional groups nV . Specifically, we begin by constructing a ranked poset P on which V G acts, and we show that the geometric realization |P| is contractible.…”

“…Our methods for analyzing the descending links are new, and are simpler than those used in [16]. Specifically, we show that the Stein complex X Stein is a simplicial subdivision of a certain complex X poly whose cells are products of simplices.…”