scholarly journals On a family of Schreier graphs of intermediate growth associated with a self-similar group

2012 ◽  
Vol 33 (7) ◽  
pp. 1408-1421 ◽  
Author(s):  
Ievgen Bondarenko ◽  
Tullio Ceccherini-Silberstein ◽  
Alfredo Donno ◽  
Volodymyr Nekrashevych
Keyword(s):  
Similar Group ◽  
Intermediate Growth ◽  
Schreier Graphs ◽  
Self Similar

"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 907-937 ◽  
Author(s):  
VADIM A. KAIMANOVICH
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Group Identity ◽  
Convex Combination ◽  
Similar Group ◽  
Grigorchuk Group ◽  
Intermediate Growth ◽  
Group A ◽  
Self Similar ◽  
Asymptotic Entropy

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.

scholarly journals A correspondence between a class of monoids and self-similar group actions II

2015 ◽  
Vol 25 (04) ◽  
pp. 633-668
Author(s):  
Mark V. Lawson ◽  
Alistair R. Wallis
Keyword(s):  
Group Actions ◽  
Similar Group ◽  
Universal Group ◽  
Length Functions ◽  
Group Of Units ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Self Similar ◽  
Cancellative Monoids

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

scholarly journals Groupoids associated to inverse semigroups

2021 ◽  
Author(s):  
Malcolm Jones
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Isomorphism ◽  
Similar Group ◽  
Self Similar

<p>Starting with an arbitrary inverse semigroup with zero, we study two well-known groupoid constructions, yielding groupoids of filters and groupoids of germs. The groupoids are endowed with topologies making them étale. We use the bisections of the étale groupoids to show there is a topological isomorphism between the groupoids. This demonstrates a widely useful equivalence between filters and germs. We use the isomorphism to characterise Exel’s tight groupoid of germs as a groupoid of filters, to find a nice basis for the topology on the groupoid of ultrafilters and to describe the ultrafilters in the inverse semigroup of an arbitrary self-similar group.</p>

scholarly journals A von Neumann algebraic approach to self-similar group actions

2019 ◽  
Vol 30 (14) ◽  
pp. 1950074
Author(s):  
Keisuke Yoshida
Keyword(s):  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Group Actions ◽  
Algebraic Approach ◽  
Similar Group ◽  
Von Neumann ◽  
Kms State ◽  
Bernoulli Measure ◽  
Self Similar ◽  
Generic Points

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

scholarly journals ON FINITE GENERATION OF SELF-SIMILAR GROUPS OF FINITE TYPE

2013 ◽  
Vol 23 (01) ◽  
pp. 69-79 ◽  
Author(s):  
IEVGEN V. BONDARENKO ◽  
IGOR O. SAMOILOVYCH
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Rooted Tree ◽  
Profinite Group ◽  
Similar Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Binary Alphabet ◽  
Self Similar

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth 3, and there are 32 such groups for depth 4.

scholarly journals FROM SELF-SIMILAR STRUCTURES TO SELF-SIMILAR GROUPS

2012 ◽  
Vol 22 (07) ◽  
pp. 1250056 ◽  
Author(s):  
DANIEL J. KELLEHER ◽  
BENJAMIN A. STEINHURST ◽  
CHUEN-MING M. WONG
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Similar Group ◽  
Limit Space ◽  
Self Similar ◽  
Necessary And Sufficient ◽  
The Relationship

We explore the relationship between the limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is post-critically finite (p.c.f.). We then give necessary and sufficient conditions for a p.c.f. self-similar structure to be the limit space of a contracting self-similar group. When these conditions hold we give a construction of the contracting group. Finally, we illustrate our results with several examples.

scholarly journals Clover nil restricted Lie algebras of quasi-linear growth

2020 ◽  
pp. 2250057
Author(s):  
Victor Petrogradsky
Keyword(s):  
Lie Algebras ◽  
Linear Growth ◽  
Positive Characteristic ◽  
Intermediate Growth ◽  
Type Formula ◽  
Periodic Groups ◽  
Restricted Lie Algebras ◽  
Self Similar ◽  
Characteristic 2 ◽  
Positive Integers

The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra 302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS) 10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras [Formula: see text], where a field of positive characteristic is arbitrary and [Formula: see text] an infinite tuple of positive integers. All these algebras have a nil [Formula: see text]-mapping. We prove that [Formula: see text]. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [Formula: see text]. We construct a subfamily of nil restricted Lie algebras [Formula: see text], with parameters [Formula: see text], [Formula: see text], having extremely slow quasi-linear growth of type: [Formula: see text], as [Formula: see text]. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2) 177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157 ]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is “almost dying” by having a “quasi-linear” growth as above, for infinitely many [Formula: see text] it has a rather fast intermediate growth of type [Formula: see text], for such periods the algebra is “resuscitating”. The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction.

AN ITERATED FUNCTION SYSTEM FOR THE CLOSURE OF THE ADDING MACHINE GROUP

Fractals ◽  
2015 ◽  
Vol 23 (03) ◽  
pp. 1550033 ◽  
Author(s):  
MUSTAFA SALTAN ◽  
BÜNYAMİN DEMİR
Keyword(s):  
Automorphism Group ◽  
Iterated Function System ◽  
Rooted Tree ◽  
Function System ◽  
Similar Group ◽  
Machine Group ◽  
Iterated Function ◽  
Self Similar ◽  
The Adding Machine

In this paper, first we equip the automorphism group of the p-ary rooted tree X* with a natural metric and define a family of contractions on Aut(X*). Then, we construct an iterated function system (IFS) whose attractor is the closure of the adding machine group on Aut(X*). Finally, we show that this group is a strong self-similar group in the sense of IFS.

Sign in / Sign up

Export Citation Format

Share Document