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

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.

"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.

Numerical upper bounds on growth of automaton groups

2021 ◽  
pp. 1-33
Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi
Keyword(s):  
Group Theory ◽  
Upper Bounds ◽  
Geometric Invariant ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Finitely Generated Group ◽  
Intermediate Growth ◽  
Mealy Automata ◽  
Optimal Bounds ◽  
Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

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

Critical microjets in collapsing cavities

1995 ◽  
Vol 290 ◽  
pp. 183-201 ◽  
Author(s):  
Michael S. Longuet-Higgins ◽  
Hasan Oguz
Keyword(s):  
Flow Field ◽  
Spherical Harmonic ◽  
Velocity Potential ◽  
Initial Conditions ◽  
Numerical Study ◽  
Critical Flow ◽  
Jet Formation ◽  
Initial Flow ◽  
The Family ◽  
Self Similar

Inward microjets are commonly observed in collapsing cavities, but here we show that jets with exceptionally high velocities and accelerations occur in certain critical flows dividing jet formation from bubble pinch-off. An example of the phenomenon occurs in the family of flows which evolve from a certain class of initial conditions: the initial flow field is that due to a moving point sink within the cavity.A numerical study of the critical flow shows that in the neighbourhood of microjet formation the flow is self-similar. The local accelerations, velocities and distances scale as tβ-2, tβ-1 and tβ respectively, where β = 0.575. The velocity potential is approximately a spherical harmonic of degree ¼.

INFRARED SCENE MODELING AND INTERPOLATION USING FRACTIONAL LÉVY STABLE MOTION

Fractals ◽  
1994 ◽  
Vol 02 (02) ◽  
pp. 303-306 ◽  
Author(s):  
STEPHEN M. KOGON ◽  
DIMITRIS G. MANOLAKIS
Keyword(s):  
Fractal Model ◽  
The Self ◽  
Dependence Structure ◽  
Fractal Interpolation ◽  
Natural Phenomena ◽  
Scene Modeling ◽  
Self Similarity ◽  
Stable Motion ◽  
The Family ◽  
Self Similar

Many data arising from natural phenomena exhibit "1/f" behavior, indicating a long-range dependence structure in the increments. The data is said to be self-similar or fractal, which has been traditionally modeled by fractional Brownian motion (fBm). This stochastic fractal model assumes a Gaussian distribution of the increments which is at times too rigid, particularly for data emanating from a long-tailed distribution. Therefore, the fractional Lévy stable motion stochastic process is proposed as a means of modeling a wider range of data. For these processes the increments are assumed to be from the family of stable distributions which have been shown to be good models of long-tailed behavior. The model is applied to data from infrared scenes and used to perform fractal interpolation, preserving not only the self-similarity, but also the probability distribution of the increments over the newly generated scales. This offers a flexible new model for a broader class of data than the fBm model.

LOWER BOUNDS ON THE GROWTH OF A GROUP ACTING ON THE BINARY ROOTED TREE

2001 ◽  
Vol 11 (01) ◽  
pp. 73-88 ◽  
Author(s):  
LAURENT BARTHOLDI
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Rooted Tree ◽  
Growth Function ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Intermediate Growth

In 1980, Rostislav Grigorchuk constructed an infinite finitely generated torsion 2-group G, called the first Grigorchuk group, and in 1983 showed that it is of intermediate growth, with the following estimates on its growth function γ (See [6]): [Formula: see text] where β= log 32(31)≈ 0.991. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to [Formula: see text] where α≈0.5157, and thus disproves the conjecture.

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.

Conjugacy growth and width of certain branch groups

2014 ◽  
Vol 24 (08) ◽  
pp. 1213-1231 ◽  
Author(s):  
Elisabeth Fink
Keyword(s):  
Lower Bound ◽  
Growth Function ◽  
Conjugacy Classes ◽  
Grigorchuk Group ◽  
Intermediate Growth ◽  
Algebraic Properties

The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius n. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further prove that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the palindromic width of some branch groups.

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