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 $K$ is collection of congruent copies of $K$ (in either Euclidean or hyperbolic space) so that no two copies intersect nontrivially in their interiors. A covering by $K$ is a collection of congruent copies of $K$ such that for every point $p$ in the space there is copy in the collection containing $p$. 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 $K$ in either $n$-dimensional Euclidean or $n$-dimensional hyperbolic space. We prove this conjecture.Comment: 14 pages, 1 figur

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

The ergodic theory of free group actions: entropy and the f-invariant

Previous work introduced two measure-conjugacy invariants: the $f$-invariant (for actions of free groups) and $\Sigma$-entropy (for actions of sofic groups). The purpose of this paper is to show that the $f$-invariant is a special case of $\Sigma$-entropy. There are two applications: the $f$-invariant is invariant under group automorphisms and there is a uniform lower bound on the $f$-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 $III_\lambda$ for some $\lambda >0$. 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 $III_0$ and, if it is weakly mixing, then it is not stable type $III_0$.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 $>1$ 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 $G$ is weakly dense in the space of actions of $G$.Comment: 16 pages. Comments welcome

A Generalization of the Prime Geodesic Theorem to Counting Conjugacy Classes of Free Subgroups

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as $x$ tends to infinity) for the number of closed geodesics with length at most $x$ on a hyperbolic manifold $M$. Closed geodesics correspond to conjugacy classes of $\pi_1(M)=\Gamma$ where $\Gamma$ is a lattice in $G=SO(n,1)$. The theorem can be rephrased in the following format. Let $X(\Z,\Gamma)$ be the space of representations of $\Z$ into $\Gamma$ modulo conjugation by $\Gamma$. $X(\Z,G)$ is defined similarly. Let $\pi: X(\Z,\Gamma)\to X(\Z,G)$ be the projection map. The PGT provides a volume form $vol$ on $X(\Z,G)$ such that for sequences of subsets $\{B_t\}$, $B_t \subset X(\Z,G)$ satisfying certain explicit hypotheses, $|\pi^{-1}(B_t)|$ is asymptotic to $vol(B_t)$. We prove a statement having a similar format in which $\Z$ is replaced by a free group of finite rank under the additional hypothesis that $n=2$ or 3.Comment: 32 pages, 5 figures. This is the second version. The introduction has been expanded and two new examples inserte