1,181 research outputs found

    The Growth of Grigorchuk's Group

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    In 1980 Rostislav Grigorchuk constructed a group GG of intermediate growth, and later obtained the following estimates on its growth function: enγ(n)enβ,e^{\sqrt{n}}\precsim\gamma(n)\precsim e^{n^\beta}, where β=log32(31)0.991\beta=\log_{32}(31)\approx0.991. Using elementary methods we bring the upper bound down to log(2)/log(2/η)0.767\log(2)/\log(2/\eta)\approx0.767, where η0.811\eta\approx0.811 is the real root of the polynomial X3+X2+X2X^3+X^2+X-2

    Regularity of aperiodic minimal subshifts

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    At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely α\alpha-repetitive, α\alpha-repulsive and α\alpha-finite (α1\alpha \geq 1), have been introduced and studied. We establish the equivalence of α\alpha-repulsive and α\alpha-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite 22-group GG. In particular, we show that these subshifts provide examples that demonstrate α\alpha-repulsive (and hence α\alpha-finite) is not equivalent to α\alpha-repetitive, for α>1\alpha > 1. We also give necessary and sufficient conditions for these subshifts to be α\alpha-repetitive, and α\alpha-repulsive (and hence α\alpha-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.Comment: 15 page

    Conjugacy Growth and Conjugacy Width of Certain Branch Groups

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    The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius nn. 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 proof 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.Comment: Final version, to appear in IJA

    Amenable groups without finitely presented amenable covers

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    The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups GG with the following properties: (i) GG is finitely generated, (ii) GG is amenable, e.g. of intermediate growth, (iii) any finitely presented group EE with a quotient isomorphic to GG contains non-abelian free subgroups, or the stronger (iii') any finitely presented group with a quotient isomorphic to GG is large

    Poisson-Furstenberg boundary and growth of groups

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    We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.Comment: 24 page
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