Entropy for expansive algebraic actions of residually finite groups (2009)
We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.
Stable orbit equivalence of Bernoulli shifts over free groups (2009)
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...
Orbit equivalence, coinduced actions and free products (2009)
The following result is proven. Let $G_1 \cc^{T_1} (X_1,\mu_1)$ and $G_2 \cc^{T_2} (X_2,\mu_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$...
The ergodic theory of free group actions: entropy and the f-invariant (2009)
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...
Weak isomorphisms between Bernoulli shifts (2008)
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.
In previous work, a measure-conjugacy invariant (called the $f$-invariant) for actions of free groups was introduced. It is analogous to the Kolmogorov-Sinai entropy. In this paper, analogues of the...
Measure conjugacy invariants for actions of countable sofic groups (2008)
Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by Weiss in [We00]. All residually finite groups (and hence every linear group) is sofic. The purpose of this paper is to...
A measure-conjugacy invariant for free group actions (2008)
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...
Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m ≥ 2, and prove that for all but countably many radii,...
Editor: R. de la Llave UNIQUENESS AND SYMMETRY IN PROBLEMS OF OPTIMALLY DENSE PACKINGS (2008)
Lewis Bowen, Charles Holton, Charles Radin
Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or...
Editor: R. de la Llave UNIQUENESS AND SYMMETRY IN PROBLEMS OF OPTIMALLY DENSE PACKINGS (2008)
Lewis Bowen, Charles Holton, Charles Radin
Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or...
Free Groups in Lattices (2008)
Let G be any locally compact, unimodular, metrizable group. The main result of this paper, roughly stated, is that if F
DENSEST PACKING OF EQUAL CIRCLES IN THE HYPERBOLIC PLANE (2007)
We propose a definition of density for packings of circles of fixed radius in the hyperbolic plane, and prove that for all but countably many radii, optimally dense packings must have low symmetry.
Abstract. We consider circle packings in the hyperbolic plane, by finitely many congruent circles, which maximize the number of touching pairs. We show that such a packing has all of its centers...
Optimally dense packings of hyperbolic space (2007)
In previous work a probabilistic approach to controlling di#culties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
DENSEST PACKING OF EQUAL CIRCLES IN THE HYPERBOLIC PLANE (2007)
We propose a definition of density for packings of circles of fixed radius in the hyperbolic plane, and prove that for all but countably many radii, optimally dense packings must have low symmetry.
Uniqueness and Symmetry in Problems of Optimally Dense Packings (2007)
Lewis Bowen, Charles Holton, Charles Radin, Lorenzo Sadun
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples...
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...
A Solidification Phenomenon in Random Packings (2005)
Bowen, Lewis, Lyons, Russell, Radin, Charles, Winkler, Peter
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.
Fluid/solid transition in a hard-core system (2005)
Bowen, Lewis, Lyons, Russell, Radin, Charles, Winkler, Peter
We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition.
Immersions of Pants into a Fixed Hyperbolic Surface (2005)
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...
Uniqueness and symmetry in problems of optimally dense packings (2005)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
Part of Hilbert's eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We...
Uniqueness and symmetry in problems of optimally dense packings (2005)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
Part of Hilbert's eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We...
Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture (2004)
We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon)...
The Gromov Norm of the Product of Two Surfaces (2004)
Bowen, Lewis, De Loera, Jesus A., Develin, Mike, Santos, Francisco
We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products...
An Isometry Between Measure Homology and Singular Homology (2004)
In Thurston's notes, he gives two different definitions of the Gromov norm (also called simplicial volume) of a manifold and states that they are equal but does not prove it. Gromov proves it in the...
Optimally Dense Packings of Hyperbolic Space (2004)
In previous work a probabilistic approach to controlling di#culties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
Sphere Packings in Hyperbolic Space: Periodicity and Continuity (2003)
This paper is being withdrawn because an error was discovered in lemma 4.3. Although the rest of the paper appears to be correct, this error invalidates the proof of theorem 3.1 and theorem 3.3.
Periodicity and Circle Packing in the Hyperbolic Plane (2003)
We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all...
Uniqueness and Symmetry in Problems of Optimally Dense Packings (2003)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples...
Couplings of Uniform Spanniing Forests (2003)
We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on Cayley graphs of finitely generated residually amenable groups.
Optimally dense packings of hyperbolic space (2002)
In previous work a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
Abstract. In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we...
Abstract. In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we...
On the existence of completely saturated packings and completely reduced covering (2001)
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...