4 research outputs found
Nilprogressions and groups with moderate growth
We show that doubling at some large scale in a Cayley graph implies uniform
doubling at all subsequent scales. The proof is based on the structure theorem
for approximate subgroups proved by Green, Tao and the first author. We also
give a number of applications to the geometry and spectrum of finite Cayley
graphs. For example, we show that a finite group has moderate growth in the
sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a
fixed power of the cardinality of the group. We call such groups almost flat
and show that they have a subgroup of bounded index admitting a cyclic quotient
of comparable diameter. We also give bounds on the Cheeger constant, first
eigenvalue of the Laplacian, and mixing time. This can be seen as a
finite-group version of Gromov's theorem on groups with polynomial growth. It
also improves on a result of Lackenby regarding property (tau) in towers of
coverings. Another consequence is a universal upper bound on the diameter of
all finite simple groups, independent of the CFSG.Comment: 37 pages. Minor changes made by a copy editor. To appear in Adv. Mat
Horofunctions on graphs of linear growth
We prove that a linear growth graph has finitely many horofunctions. This
provides a short and simple proof that any finitely generated infinite group of
linear growth is virtually cyclic
Amenability of groups and -sets
This text surveys classical and recent results in the field of amenability of
groups, from a combinatorial standpoint. It has served as the support of
courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure.
The goals of the text are (1) to be as self-contained as possible, so as to
serve as a good introduction for newcomers to the field; (2) to stress the use
of combinatorial tools, in collaboration with functional analysis, probability
etc., with discrete groups in focus; (3) to consider from the beginning the
more general notion of amenable actions; (4) to describe recent classes of
examples, and in particular groups acting on Cantor sets and topological full
groups