This paper is motivated by an old and central problem in measurable dynamics: given two dynamical systems, determine whether or not they are measurably conjugate, i.e., isomorphic. Let us set some… (More)

We prove that given a fixed radius r, the maximum density achieved by packings of the hyperbolic plane by radius r circles is the supremum of densities of “periodic packings” (those packings with… (More)

This paper introduces Markov chains and processes over nonabelian free groups and semigroups. We prove a formula for the f -invariant of a Markov chain over a free group in terms of transition… (More)

Previous work introduced two measure-conjugacy invariants: the f-invariant (for actions of free groups) and Σ-entropy (for actions of sofic groups). The purpose of this paper is to show that the… (More)

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.

We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.

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… (More)

We prove that uniformly random packings of copies of a certain simply-connected figure in the plane exhibit global connectedness at all sufficiently high densities, but not at low densities.

Let G be any locally compact metrizable group. The main result of this paper, roughly stated, is that if F < G is any finitely generated free group and Γ < G any lattice, then up to a small… (More)

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if G 1 , G 2 are nonabelian free groups of finite rank… (More)