Metric properties of braided Thompson's groups

Puig, Josep Burillo; Cleary, Sean

doi:10.1512/iumj.2009.58.3468

Cited by 11 publications

(5 citation statements)

References 8 publications

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“…This is closely related to the strand diagram model for Thompson's groups in [BM13]. See also Section 1.2 in [Bri07] and Figure 2 of [BC09].…”

Section: The Braided Thompson's Groupssupporting

confidence: 55%

The braided Thompson's groups are of type $F_\infty$

Bux,

Fluch,

Marschler

et al. 2012

Preprint

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We prove that the braided Thompson's groups V br and F br are of type F∞, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected. In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.A group is of type F ∞ if it admits a classifying space whose n-skeleton is compact for every n. The case n = 2 shows that such a group is in particular finitely presented. Prominent examples of groups of type F ∞ include Thompson's groups F , T and V , and the braid groups B n . A braided variant of Thompson's group V , which we will denote V br , was introduced independently by Brin and Dehornoy [Bri07,Deh06]. This group contains F as a subgroup, along with copies of the braid group B n for each n ∈ N, and was shown to be finitely presented by Brin [Bri06]. Brady, Burillo, Cleary and Stein [BBCS08] introduced another braided Thompson's group, which we denote F br , and which contains the pure braid groups P B n in a similar way to how V br contains B n . They also proved that F br is finitely presented. The notation used in [Bri07, BBCS08] is BV and BF . The relationship between V and V br is in many ways reminiscent of the relationship between a Coxeter group and its corresponding Artin group. For example, there is a presentation of V that can be converted to a presentation for V br by dropping the relations that the generators are involutions [Bri06].In this paper we prove that the braided Thompson's groups are of type F ∞ . In the case of V br , this was conjectured by John Meier already in 2001. This question was also discussed in [FK08, Remark 5.1 (1)] and [FK11, Remark 3.3].Main Theorem. The braided Thompson's groups V br and F br are of type F ∞ .Our proof is geometric. The starting point is that each braided Thompson's group acts naturally on an associated poset complex. One key step is to restrict this action to an invariant cubical subcomplex that is smaller and therefore easier

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“…This is closely related to the strand diagram model for Thompson's groups in [BM13]. See also Section 1.2 in [Bri07] and Figure 2 of [BC09].…”

Section: The Braided Thompson's Groupssupporting

confidence: 55%

The braided Thompson's groups are of type $F_\infty$

Bux,

Fluch,

Marschler

et al. 2012

Preprint

View full text Add to dashboard Cite

show abstract

“…Like V , it is finitely presented [Bri06] and of type F ∞ [BFM + 16]. Further results include an inspection of metric properties of V br [BC09] and normal subgroups [Zar]. In particular, while V br is not simple like V is, since it surjects onto V much like B n surjects onto S n , it is true that every proper normal subgroup of V br lies in the kernel of this quotient to V [Zar].…”

Section: Braided Thompson Groupsmentioning

confidence: 99%

On normal subgroups of the braided Thompson groups

Zaremsky

2018

Groups Geom. Dyn.

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In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type F ∞ . The proof utilized certain cube complexes with a Morse function, whose descending links were modeled by complexes of arcs on surfaces called matching arc complexes. In this paper we establish some geometric results, namely that the cube complexes are CAT(0) and the so called complete matching arc complexes are δ-hyperbolic. We then compute the geometric invariants Σ m (F br ) of the pure braided Thompson group F br . Only the first invariant Σ 1 (F br ) was previously known. A consequence of our computation is that as soon as a subgroup of F br containing the commutator subgroup [F br , F br ] is finitely presented, it is automatically of type F ∞ .

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“…Other braided Thompson groups include the "F -like" pure braided Thompson groups bF [12], various "T -like" braided Thompson groups [33,34,59], braided Higman-Thompson groups bV n [4,53], braided Brin-Thompson groups sV br [56], the "ribbon braided" Thompson group rV [57] and braided Röver-Nekrashevych groups brV d (G) [55]. Most relevant to our purposes here is a close relative bV of bV , which was also introduced by Brin in [14] (there denoted BV ), and realised up to isomorphism as a concrete subgroup of bV by Brady-Burillo-Cleary-Stein [12]; see also [18].…”

Section: Braided Thompson Groupsmentioning

confidence: 99%

Braided Thompson groups with and without quasimorphisms

Fournier-Facio¹,

Lodha²,

Zaremsky³

2022

Preprint

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We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin-Dehornoy braided Thompson group bV has an infinite-dimensional space of quasimorphisms and thus infinite-dimensional second bounded cohomology. This implies that despite being perfect, bV is not uniformly perfect, in contrast to Thompson's group V . We also prove that relatives of bV like the ribbon braided Thompson group rV and the pure braided Thompson group bF similarly have an infinite-dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of bV denoted bV , which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes bV the first example of a left-orderable group of type F ∞ that is not locally indicable and has trivial second bounded cohomology. This also makes bV an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is non-amenable but has trivial second bounded cohomology, behaviour that cannot happen for finite-type mapping class groups.

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Metric properties of braided Thompson's groups

Cited by 11 publications

References 8 publications

The braided Thompson's groups are of type $F_\infty$

The braided Thompson's groups are of type $F_\infty$

On normal subgroups of the braided Thompson groups

Braided Thompson groups with and without quasimorphisms

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