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

Belk, James; Matucci, Francesco

doi:10.5565/publmat_60216_07

Cited by 11 publications

(59 citation statements)

References 15 publications

(25 reference statements)

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“…The complex X (S * ≀ G) H is still not quite nice enough to get higher connectivity of descending links. In this subsection, following [BM16] we will glom certain simplices in X (S * ≀ G) H together to reveal a coarser, polysimplicial cell structure.…”

Section: The H-stein-farley Complexmentioning

confidence: 99%

“…For the case when G is not of type F ∞ , but some orderly A-coarsely self-similar subgroup H that is nuclear in G is of type F ∞ , it is much less clear whether one could apply Thumann's approach. We suspect not, since it seems like Thumann's framework would demand that the stabilizers be built out of G rather than getting to choose a nice H. One could probably modify Thumann's machine to handle these groups, by first translating the approach here and in [BM16] into the language of operads. It would be interesting to see if using operads could yield F ∞ for any Nekrashevych groups beyond the ones handled here.…”

Section: Descending Linksmentioning

confidence: 99%

“…There is evidence that as soon as G is finitely generated and contracting, V d (G) is of type F ∞ . Nekrashevych proved that such V d (G) are at least finitely presentable [Nek13, Theorem 5.9], and according to the last paragraph in the introduction of [BM16] Bartholdi and Geoghegan have some unpublished results proving that they are indeed…”

Section: Almost-automorphisms Of Trees Cloning Systems and Finitenesmentioning

confidence: 99%

“…Acknowledgments. We are grateful to Jim Belk and Francesco Matucci for supplying some helpful information about their paper [BM16].…”

mentioning

confidence: 99%

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Almost-automorphisms of trees, cloning systems and finiteness properties

Skipper

Zaremsky

2019

J. Topol. Anal.

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We prove that the group of almost-automorphisms of the infinite rooted regular d-ary tree T d arises naturally as the Thompson-like group of a so called d-ary cloning system. A similar phenomenon occurs for any Nekrashevych group V d (G), for G ≤ Aut(T d ) a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for G, is of type F ∞ . Namely, we find some natural conditions on subgroups of G to ensure that V d (G) is of type F ∞ , and in particular we prove this for all G in the infinite family of Sunić groups. We also prove that if G is itself of type F ∞ then so is V d (G), and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group G yields a type F ∞ Nekrashevych group V d (G).

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Section: The H-stein-farley Complexmentioning

confidence: 99%

Section: Descending Linksmentioning

confidence: 99%

Section: Almost-automorphisms Of Trees Cloning Systems and Finitenesmentioning

confidence: 99%

“…Acknowledgments. We are grateful to Jim Belk and Francesco Matucci for supplying some helpful information about their paper [BM16].…”

mentioning

confidence: 99%

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Almost-automorphisms of trees, cloning systems and finiteness properties

Skipper

Zaremsky

2019

J. Topol. Anal.

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“…We can mention the papers of [17,16,8,1,22] which all explore properties of such groups. Our own perspective has been heavily influenced by the dynamical methods which have arisen through the use of Rubin's theorem [15,18] and, as will be seen below, through the interaction between the theory of the extended Thompson type groups and the theory of groups of automata (c.f., [11]).…”

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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Simple groups separated by finiteness properties

2018

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We show that for every positive integer n there exists a simple group that is of type F n−1 but not of type F n . For n ≥ 3 these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace-Rémy, consists of non-affine Kac-Moody groups over finite fields. Our examples arise from Röver-Nekrashevych groups, and contain free abelian groups of infinite rank.

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

Cited by 11 publications

References 15 publications

Almost-automorphisms of trees, cloning systems and finiteness properties

Almost-automorphisms of trees, cloning systems and finiteness properties

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

Simple groups separated by finiteness properties

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