996 research outputs found
Power-free values of the polynomial t_1...t_r - 1
Let k, r > 1 be two integers. We prove an asymptotic formula for the number
of k-free values of the r variables polynomial t_1...t_r - 1 over the integral
points of [1, x]^r.Comment: Final versio
Groups acting on trees with almost prescribed local action
We investigate a family of groups acting on a regular tree, defined by
prescribing the local action almost everywhere. We study lattices in these
groups and give examples of compactly generated simple groups of finite
asymptotic dimension (actually one) not containing lattices. We also obtain
examples of simple groups with simple lattices, and we prove the existence of
(infinitely many) finitely generated simple groups of asymptotic dimension one.
We also prove various properties of these groups, including the existence of a
proper action on a CAT(0) cube complex.Comment: v2: 35 pages; argument slightly modified in 4.2.2; final versio
Compact presentability of tree almost automorphism groups
We establish compact presentability, i.e. the locally compact version of
finite presentability, for an infinite family of tree almost automorphism
groups. Examples covered by our results include Neretin's group of
spheromorphisms, as well as the topologically simple group containing the
profinite completion of the Grigorchuk group constructed by Barnea, Ershov and
Weigel.
We additionally obtain an upper bound on the Dehn function of these groups in
terms of the Dehn function of an embedded Higman-Thompson group. This, combined
with a result of Guba, implies that the Dehn function of the Neretin group of
the regular trivalent tree is polynomially bounded.Comment: The results are extended to some almost automorphism groups of trees
associated with closed regular branch groups. In particular we prove that the
simple group (containing the profinite completion of the Grigorchuk group)
constructed by Barnea, Ershov and Weigel, is compactly presente
The Stationary Behaviour of Fluid Limits of Reversible Processes is Concentrated on Stationary Points
Assume that a stochastic processes can be approximated, when some scale
parameter gets large, by a fluid limit (also called "mean field limit", or
"hydrodynamic limit"). A common practice, often called the "fixed point
approximation" consists in approximating the stationary behaviour of the
stochastic process by the stationary points of the fluid limit. It is known
that this may be incorrect in general, as the stationary behaviour of the fluid
limit may not be described by its stationary points. We show however that, if
the stochastic process is reversible, the fixed point approximation is indeed
valid. More precisely, we assume that the stochastic process converges to the
fluid limit in distribution (hence in probability) at every fixed point in
time. This assumption is very weak and holds for a large family of processes,
among which many mean field and other interaction models. We show that the
reversibility of the stochastic process implies that any limit point of its
stationary distribution is concentrated on stationary points of the fluid
limit. If the fluid limit has a unique stationary point, it is an approximation
of the stationary distribution of the stochastic process.Comment: 7 pages, preprin
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