28 research outputs found
Imbeddings into groups of intermediate growth
Every countable group that does not contain a finitely generated subgroup of
exponential growth imbeds in a finitely generated group of subexponential
growth.
This produces in particular the first examples of groups of subexponential
growth containing the additive group of the rationals.Comment: Compared to v1, some typos corrected, and the part on distortion
split off to arXiv:1406.590
Isoperimetric inequalities, shapes of F{\o}lner sets and groups with Shalom's property
We prove an isoperimetric inequality for groups. As an application, we obtain
lower bound on F{\o}lner functions in various nilpotent-by-cyclic groups. Under
a regularity assumption, we obtain a characterization of F{\o}lner functions of
these groups. As another application, we evaluate the asymptotics of the
F{\o}lner function of . We construct new
examples of groups with Shalom's property , in particular
among nilpotent-by-cyclic and lacunary hyperbolic groups. Among these examples
we find groups with property , which are direct products of
lacunary hyperbolic groups and have arbitrarily large F{\o}lner functions
Groups of given intermediate word growth
We show that there exists a finitely generated group of growth ~f for all
functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq
f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of
X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than
\exp(R^{\log2/\log\eta}).
We also give a family of self-similar branched groups of growth
~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).Comment: small typos corrected from v
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
Distortion of imbeddings of groups of intermediate growth into metric spaces
For every metric space in which there exists a sequence of
finite groups of bounded-size generating set that does not embed coarsely, and
for every unbounded, increasing function , we produce a group of
subexponential word growth all of whose imbeddings in have
distortion worse than .
This applies in particular to any B-convex Banach space , such as
Hilbert space.Comment: Used to appear as first half of arXiv:1403.558