Local similarities and the Haagerup property (with an appendix by Daniel S. Farley)

Hughes, Bruce

doi:10.4171/ggd/58

Cited by 18 publications

(26 citation statements)

References 14 publications

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“…We begin with a review of finite similarity structures on compact, ultrametric spaces, as defined in Hughes [10].…”

Section: Groups Defined By Finite Similarity Structuresmentioning

confidence: 99%

“…The results of [10] left many open questions about the new class of FSS groups. In this paper, guided by previous work on Thompson's group V and related groups, we will establish several new properties of FSS groups.…”

mentioning

confidence: 99%

“…Hughes [10] proved that all FSS groups have the Haagerup property. His argument even established the stronger conclusion that all FSS groups act properly by isometries on CAT(0) cubical complexes.…”

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confidence: 99%

“…In [10], Hughes defined a class of groups that act by homeomorphisms on compact ultrametric spaces. Let X be a compact ultrametric space.…”

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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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“…We begin with a review of finite similarity structures on compact, ultrametric spaces, as defined in Hughes [10].…”

Section: Groups Defined By Finite Similarity Structuresmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…In [10], Hughes defined a class of groups that act by homeomorphisms on compact ultrametric spaces. Let X be a compact ultrametric space.…”

mentioning

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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“…Elizabeth Scott in her research [19,20,21] describes the first such family of groups which are developed further by Claas Röver, specifically including an extension of V with the Grigorchuk group Γ. The second family are the finite similarity structure groups of Hughes (FSS groups), which are the focus of study in [14,8].…”

Section: Introductionmentioning

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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Operad groups and their finiteness properties

Thumann

2017

Advances in Mathematics

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Abstract. We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type F∞. This unifies and extends existing proofs that certain Thompson-like groups are of type F∞.

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Local similarities and the Haagerup property (with an appendix by Daniel S. Farley)

Cited by 18 publications

References 14 publications

Finiteness Properties of Some Groups of Local Similarities

Finiteness Properties of Some Groups of Local Similarities

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

Operad groups and their finiteness properties

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