1,181 research outputs found
The Growth of Grigorchuk's Group
In 1980 Rostislav Grigorchuk constructed a group of intermediate growth,
and later obtained the following estimates on its growth function:
where
. Using elementary methods we bring the upper
bound down to , where is
the real root of the polynomial
From self-similar groups to self-similar sets and spectra
The survey presents developments in the theory of self-similar groups leading
to applications to the study of fractal sets and graphs, and their associated
spectra
Regularity of aperiodic minimal subshifts
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 -repetitive,
-repulsive and -finite (), have been introduced
and studied. We establish the equivalence of -repulsive and
-finite for general subshifts over finite alphabets. Further, we
studied a family of aperiodic minimal subshifts stemming from Grigorchuk's
infinite -group . In particular, we show that these subshifts provide
examples that demonstrate -repulsive (and hence -finite) is not
equivalent to -repetitive, for . We also give necessary and
sufficient conditions for these subshifts to be -repetitive, and
-repulsive (and hence -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
The conjugacy growth function counts the number of distinct conjugacy classes
in a ball of radius . 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
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 with the following properties: (i) is finitely generated, (ii)
is amenable, e.g. of intermediate growth, (iii) any finitely presented
group with a quotient isomorphic to contains non-abelian free
subgroups, or the stronger (iii') any finitely presented group with a quotient
isomorphic to is large
Poisson-Furstenberg boundary and growth of groups
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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