Factorizations of Self-similar Groups Associated to the First Grigorchuk Group

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

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COSET ENUMERATION FOR CERTAIN INFINITELY PRESENTED GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006637 ◽

2011 ◽

Vol 21 (08) ◽

pp. 1369-1380 ◽

Cited By ~ 2

Author(s):

RENÉ HARTUNG

Keyword(s):

Finite Index ◽

Membership Problem ◽

Finitely Generated ◽

Grigorchuk Group ◽

Self Similar ◽

Coset Enumeration ◽

Finite Index Subgroups ◽

Subgroup Membership Problem

We describe an algorithm that computes the index of a finitely generated subgroup in a finitely L-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely L-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group, and the Hanoi 3-group.

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Generalized Grigorchuk’s overgroups as points in the space of marked 8-generated groups

Journal of Algebra and Its Applications ◽

10.1142/s021949882250058x ◽

2020 ◽

pp. 2250058

Author(s):

Supun T. Samarakoon

Keyword(s):

Grigorchuk Group ◽

Intermediate Growth ◽

The Family ◽

Self Similar

First Grigorchuk group [Formula: see text] and Grigorchuk’s overgroup [Formula: see text], introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, [Formula: see text] was used to construct the family of generalized Grigorchuk groups [Formula: see text], which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup [Formula: see text] to the family [Formula: see text] of generalized Grigorchuk’s overgroups. We consider these groups as 8-generated and describe the closure of this family in the space [Formula: see text] of marked [Formula: see text]-generated groups.

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