Embeddings into Thompson's group
            <i>V</i>
            and coCF groups

Bleak, Collin; Matucci, Francesco; Neunhöffer, Max

doi:10.1112/jlms/jdw044

Cited by 26 publications

(64 citation statements)

References 20 publications

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“…In our next application, we extend [BMN16,Theorem 4] where it is proved that QV 2,1,0 embeds into V .…”

Section: Applicationsmentioning

confidence: 87%

Embeddings into Thompson's groups from quasi-median geometry

Genevois

2019

Groups Geom. Dyn.

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The main result of this article is that any braided (resp. annular, planar) diagram group D splits as a short exact sequence 1 → R → D → S → 1 where R is a subgroup of some right-angled Artin group and S a subgroup of Thompson's group V (resp. T , F ). As an application, we show that several braided diagram groups embeds into Thompson's group V , including Higman's groups Vn,r, groups of quasi-automorphisms QVn,r,p, and generalised Houghton's groups Hn,p. arXiv:1709.03888v2 [math.GR]

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“…In our next application, we extend [BMN16,Theorem 4] where it is proved that QV 2,1,0 embeds into V .…”

Section: Applicationsmentioning

confidence: 87%

Embeddings into Thompson's groups from quasi-median geometry

Genevois

2019

Groups Geom. Dyn.

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“…One cause for such interest comes from a conjecture of Lehnert, modified by Bleak, Matucci, and Neuhöffer in [5], that Thompson's group V is a universal group with context-free co-word problem, i.e. a universal coC F group (so every finitely generated subgroup of V is a coC F group, and all coC F groups embed into V ).…”

Section: N=4mentioning

confidence: 99%

Some isomorphism results for Thompson-like groups V n (G)

Bleak¹,

Donoven²,

Jonušas³

2017

Isr. J. Math.

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Abstract. In this paper, we provide some isomorphism results for the groups {Vn(G)}, supergroups of the Higman-Thompson group Vn where n ∈ N and G ≤ Sn, the symmetric group on n points. These groups, introduced by Farley and Hughes, are the groups generated by Vn and the tree automorphisms [α]g defined as follows. For each g ∈ G and each node α in the infinite rooted n-ary tree, the automorphisms [α]g acts iteratively as g on the child leaves of α and every descendent of α. In particular, we show that Vn ∼ = Vn(G) if and only if G is semiregular (acts freely on n points) and some additional sufficient conditions for isomorphisms. Essential tools in the above work are a study of the dynamics of the action of elements of Vn(G) on the Cantor space, Rubin's Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiȋ's rational group on the n-ary alphabet.

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“…Section 5 sketches some possible further developments, including sketches of the proofs thatQT andQV have type F ∞ (as already proved by [14]), and the other generalizations briefly described above. 2. Braided diagram groups and actions on associated complexes 2.…”

Section: Introductionmentioning

confidence: 99%

“…We say that ∆ is annular if it can be similarly embedded in an annulus. Or, more precisely, suppose that we replace the frame ∂([0, 1] 2 ) with a pair of disjoint circles, each of which is given the standard counterclockwise orientation, in place of the usual left-right orientations on the top and bottom of ∂([0, 1]) 2 . We further give both circles basepoints, which are to be disjoint from all contacts.…”

Section: Introductionmentioning

confidence: 99%

Quasiautomorphism groups of type F∞

Audino¹,

Aydel

Farley

2018

Algebr. Geom. Topol.

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The groups QF , QT ,QT ,QV , and QV are groups of quasiautomorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F , T , and V .We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F∞. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.2010 Mathematics Subject Classification. 20F65, 57M07.The third author showed (in [11]) that QV is a braided diagram group over a semigroup presentation. This description suggests an approach to proving the F ∞ property for QV . Since [10] shows that a class of braided diagram groups (including Thompson's group V ) have type F ∞ , and in fact other classes of diagram groups were shown to have type F ∞ in [8] and [10], it is at least plausible that some approach inspired by the theory of diagram groups could establish the F ∞ property for QV and QT . (We note that the original proofs that Thompson's groups F , T , and V have type F ∞ were given by Brown [4] and by Brown and Geoghegan [5] in the 1980s.)Nucinkis and St. John-Green show, however, that the hypotheses of the main theorem in [10] are satisfied by neither QT nor QV . In fact, as also noted in [14], even the much more general main theorem of [15] does not apply to either of QT and QV .The goal of the present article is to extend the diagram-group methods of [8] and [10] to the groups QF , QT , QV ,QT , andQV . We will show that all of these groups can be described using the theory of diagram groups over semigroup presentations. Indeed, all of these groups are diagram groups over the same semigroup presentation, namely P = x, a | x = xax , although the specific types of diagram vary from group to group. Three types of diagram groups have been considered in the literature: planar, annular, and braided diagram groups. All were introduced by Guba and Sapir in [12], which devotes by far the greatest attention to planar diagram groups (which are usually simply called diagram groups). The papers [9] and [10] consider the annular and braided diagram groups in more detail. Here we will introduce hybrid diagram groups that combine properties of multiple diagram group types. For instance, the group QF is a special type of diagram group over P, in which the diagrams exhibit both planar and braided behavior at the same time. We will use such hybrid diagrams to prove that the groups QF , QT , QV ,QT , and QV all act properly by isometries on CAT(0) cubical complexes, and that all have type F ∞ . In fact, our methods extend with equal ease to the case of an arbitrary finite number of binary trees and isolated vertices (see Section 5), and the case of n-ary trees (for fixed n ≥ 2) is different only in the details. It even seems likely that our argument generalizes to other, non-regular, trees, although this i...

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Embeddings into Thompson's group V and coCF groups

Cited by 26 publications

References 20 publications

Embeddings into Thompson's groups from quasi-median geometry

Embeddings into Thompson's groups from quasi-median geometry

Some isomorphism results for Thompson-like groups V n (G)

Quasiautomorphism groups of type F∞

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