Higher Dimensional Thompson Groups

Brin, Matthew G.

doi:10.1007/s10711-004-8122-9

Cited by 84 publications

(160 citation statements)

References 17 publications

(37 reference statements)

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“…We determine the abelianization of V ′ d (H), show that it is always finite, and prove that the commutator subgroup [V ′ d (H), V ′ d (H)] is simple. Our arguments draw on work of Nekrashevych [11] and Brin [3].…”

Section: Theorem 11 (Main Theoremmentioning

confidence: 99%

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Finiteness Properties of Some Groups of Local Similarities

Farley

Hughes

2015

Proceedings of the Edinburgh Mathematical Society

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Abstract. Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson's group V .Guided by previous work on Thompson's group V , we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F∞. This generalizes work of Ken Brown from the 1980s. Next, we develop methods for distinguishing between isomorphism types of some of the Nekrashevych-Röver groups V d (H), where H is a finite group, and show that all such groups V d (H) have simple subgroups of finite index. Lastly, we show that FSS groups defined by small Sim-structures are braided diagram groups over tree-like semigroup presentations. This generalizes a result of Guba and Sapir, who first showed that Thompson's group V is a braided diagram group.

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Section: Theorem 11 (Main Theoremmentioning

confidence: 99%

“…Brin [3] used the first part of the following corollary to show that 2V and V are not isomorphic. Bleak and Lanoue [2] used the second part to show nV is not isomorphic to mV if n = m. (1) For every f ∈ F and for every x ∈ X, h induces a bijection from the orbit {f n x | n ∈ Z} to the orbit {(φ(f )) n (h(x)) | n ∈ Z}.…”

Section: Theorem 72 (Rubin)mentioning

confidence: 99%

Finiteness Properties of Some Groups of Local Similarities

Farley

Hughes

2015

Proceedings of the Edinburgh Mathematical Society

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“…When n is odd, one must pass to the commutator subgroup of index 2, reflecting the observation that the corresponding split relations in G n,r do not change the parity of any decomposition as a product of transpositions.) Other families include the Brin-Thompson groups nV for which V = 1V , see [6], and the groups nV m,r that generalise the previous two families, see [20], and where we have similar simplicity considerations, see [7]. The finite presentability of these groups comes from the much stronger fact that they are all in fact F ∞ groups.…”

Section: Introductionmentioning

confidence: 95%

“…Since then, it has been the focus of a large amount of subsequent research (see, for example, [4,5,6,13,15,18,24] for a small part of that research). Thompson's group V arises in various other settings, for example, Birget [2,3] investigates connections to circuits and complexity while Lawson [19] considers links to inverse monoids and etale groupoids.…”

Section: Introductionmentioning

confidence: 99%

The infinite simple group $V$ of Richard J. Thompson: presentations by permutations

Bleak¹,

Quick²

2017

Groups Geom. Dyn.

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We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group V , known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation and which enables us to develop two small presentations for V : a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.Mathematics Subject Classification (2010). Primary: 20F05; Secondary: 20E32, 20F65.

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“…It is then shown that the geometric realisation of this poset yields the required finiteness properties. In [6] M. Brin defined a group sV generalising V for every natural number s ≥ 2. Analogously to V , these groups are defined as subgroups of the homeomorphism group of a finite Cartesian product of the Cantor-set.…”

Section: Introductionmentioning

confidence: 99%