Constructing Finitely Presented Simple Groups That Contain Grigorchuk Groups

Röver, Claas E.

doi:10.1006/jabr.1999.7898

Cited by 39 publications

(38 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We present many results without proof, but in most cases the proofs can be found in either [24] (for results on G), [9] (for results on V ), or [26] (for results on V G)…”

Section: Notation and Backgroundmentioning

confidence: 99%

See 1 more Smart Citation

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

Belk¹,

Matucci²

2016

Publicacions Matemàtiques

View full text Add to dashboard Cite

Abstract. We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimiplical complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

show abstract

“…We present many results without proof, but in most cases the proofs can be found in either [24] (for results on G), [9] (for results on V ), or [26] (for results on V G)…”

Section: Notation and Backgroundmentioning

confidence: 99%

“…Let V G be the group of homeomorphisms of the Cantor set generated by Thompson's group V and Grigorchuk's first group G. This group was considered by Claas Röver, who proved that V G is finitely presented and simple [26], and also that V G is isomorphic to the abstract commensurator of G [27].…”

Section: Introductionmentioning

confidence: 99%

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

Belk¹,

Matucci²

2016

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…The group generated by V 2 and the Grigorchuk group was studied by Roever [40]. He proved that it is a finitely presented simple group isomorphic to the abstract commensurizer of the Grigorchuk group.…”

Section: Proposition 425mentioning

confidence: 99%

Self-similar groups, automatic sequences, and unitriangular representations

et al. 2015

View full text Add to dashboard Cite

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two separate notions of automaticity: groups generated by automata and automatic sequences. We also show that if the group acts on the tree by p-adic automorphisms, then the corresponding linear representation is a representation by infinite triangular matrices. We relate this observation with the notion of height of an automorphism of a rooted tree due to L. Kaloujnine.

show abstract

“…! / generalizing a construction of Röver [19]. To describe these groups note that there is a natural similarity from A !…”

Section: Example 43 (Subgroups)mentioning

confidence: 99%

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

Hughes¹

2009

Groups Geom. Dyn.

View full text Add to dashboard Cite

Abstract.A new class of groups, the locally finitely determined groups of local similarities on compact ultrametric spaces, is introduced and it is proved that these groups have the Haagerup property (that is, they are a-T-menable in the sense of Gromov). The class includes Thompson's groups, which have already been shown to have the Haagerup property by D. S. Farley, as well as many other groups acting on boundaries of trees. A sufficient condition, used in this article, for the Haagerup property is shown in the appendix by D. S. Farley to be equivalent to the well-known property of having a proper action on a space with walls. Mathematics Subject Classification (2000). 20F65, 22D10, 54E45.

show abstract

Constructing Finitely Presented Simple Groups That Contain Grigorchuk Groups

Abstract: We construct infinite finitely presented simple groups that have subgroups isomorphic to Grigorchuk groups. We also prove that up to one possible exception all previously known finitely presented simple groups are torsion locally finite.

Cited by 39 publications

References 9 publications

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

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

Self-similar groups, automatic sequences, and unitriangular representations

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

Contact Info

Product

Resources

About