2,608 research outputs found
Weak isomorphisms between Bernoulli shifts
In this note, we prove that if G is a countable group that contains a
nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G
are weakly isomorphic.Comment: 10 page
Equivalence relations that act on bundles of hyperbolic spaces
Consider a measured equivalence relation acting on a bundle of hyperbolic
metric spaces by isometries. We prove that every aperiodic hyperfinite
subequivalence relation is contained in a {\em unique} maximal hyperfinite
subequivalence relation. We classify elements of the full group according to
their action on fields on boundary measures (extending earlier results of
Kaimanovich), study the existence and residuality of different types of
elements and obtain an analogue of Tits' alternative.Comment: this version is the final versio
Random walks on random coset spaces with applications to Furstenberg entropy
We determine the range of Furstenberg entropy for stationary ergodic actions
of nonabelian free groups by an explicit construction involving random walks on
random coset spaces.Comment: A few minor corrections have been made since the previous versio
On the existence of completely saturated packings and completely reduced covering
A packing by a body is collection of congruent copies of (in either
Euclidean or hyperbolic space) so that no two copies intersect nontrivially in
their interiors. A covering by is a collection of congruent copies of
such that for every point in the space there is copy in the collection
containing . A completely saturated packing is one in which it is not
possible to replace a finite number of bodies of the packing with a larger
number and still remain a packing. A completely reduced covering is one in
which it is not possible to replace a finite number of bodies of the covering
with a smaller number and still remain a covering. It was conjectured by G.
Fejes Toth, G. Kuperberg, and W. Kuperberg that completely saturated packings
and commpletely reduced coverings exist for every body in either
-dimensional Euclidean or -dimensional hyperbolic space. We prove this
conjecture.Comment: 14 pages, 1 figur
The ergodic theory of free group actions: entropy and the f-invariant
Previous work introduced two measure-conjugacy invariants: the -invariant
(for actions of free groups) and -entropy (for actions of sofic
groups). The purpose of this paper is to show that the -invariant is a
special case of -entropy. There are two applications: the -invariant
is invariant under group automorphisms and there is a uniform lower bound on
the -invariant of a factor in terms of the original system.Comment: 14 pages. This version corrects minor error
The type and stable type of the boundary of a Gromov hyperbolic group
Consider an ergodic non-singular action \Gamma \cc B of a countable group
on a probability space. The type of this action codes the asymptotic range of
the Radon-Nikodym derivative, also called the {\em ratio set}. If \Gamma \cc
X is a pmp (probability-measure-preserving) action, then the ratio set of the
product action \Gamma \cc B\times X is contained in the ratio set of \Gamma
\cc B. So we define the {\em stable ratio set} of \Gamma \cc B to be the
intersection over all pmp actions \Gamma \cc X of the ratio sets of \Gamma
\cc B\times X. By analogy, there is a notion of {\em stable type} which codes
the stable ratio set of \Gamma \cc B. This concept is crucially important for
the identification of the limit in pointwise ergodic theorems established by
the author and Amos Nevo.
Here, we establish a general criteria for a nonsingular action of a countable
group on a probability space to have stable type for some
. This is applied to show that the action of a non-elementary
Gromov hyperbolic group on its boundary with respect to a quasi-conformal
measure is not type and, if it is weakly mixing, then it is not stable
type .Comment: Comments welcome
Simple and large equivalence relations
We construct ergodic discrete probability measure preserving equivalence
relations \cR that has no proper ergodic normal subequivalence relations and
no proper ergodic finite-index subequivalence relations. We show that every
treeable equivalence relation satisfying a mild ergodicity condition and cost
surjects onto every countable group with ergodic kernel. Lastly, we
provide a simple characterization of normality for subequivalence relations and
an algebraic description of the quotient.Comment: Comments welcome! This new version includes expanded reference to
previous work of Stefaan Vaes which constructs equivalence relations without
finite extensions or ergodic finite index sub relations. Also I shortened the
paper by removing my construction with Pierre-Emmanuel Caprace since
Stefaan's constructions are based on the same principle
A measure-conjugacy invariant for free group actions
This paper introduces a new measure-conjugacy invariant for actions of free
groups. Using this invariant, it is shown that two Bernoulli shifts over a
finitely generated free group are measurably conjugate if and only if their
base measures have the same entropy. This answers a question of Ornstein and
Weiss.Comment: The proofs in this version are slightly simpler than in the previous
version. Also, the last 3 sections have been removed. I intend to write up
the main results of those sections in a separate pape
Weak density of orbit equivalence classes of free group actions
It is proven that the orbit-equivalence class of any essentially free
probability-measure-preserving action of a free group is weakly dense in
the space of actions of .Comment: 16 pages. Comments welcome
Immersions of Pants into a Fixed Hyperbolic Surface
Exploiting a relationship between closed geodesics on a generic closed
hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x
T_1(S), we obtain a local asymptotic equidistribution result for long closed
geodesics on S. Applications include asymptotic estimates for the number of
pants immersions into S satisfying various geometric constraints. Also we show
that two uniformly random closed geodesics gamma_1, gamma_2 of length within
epsilon of L have a high probability of partially bounding an immersed 4-holed
sphere whose other boundary components also have length within epsilon of L.
Version 2: The coefficient in corollary 1.6 corrected from 16 to 8.Comment: 43 page
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