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Lewis Bowen

Publications and source records attributed to Lewis Bowen.

At least 73 records · Page 4Linked to original sources
arXiv2009

Stable orbit equivalence of Bernoulli shifts over free groups

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 then every nontrivial Bernoulli shift over $G_1$ is stably orbit equivalent to every nontrivial Bernoulli shift over $G_2$. This answers a question of S. Popa.

math.DS↗
arXiv2009

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 $Σ$-entropy (for actions of sofic groups). The purpose of this paper is to show that the $f$-invariant is a special case of $Σ$-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.

math.DS↗
arXiv2009

Measure conjugacy invariants for actions of countable sofic groups

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 introduce, for every countable sofic group $G$, a family of measure-conjugacy invariants for measure-preserving $G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property T groups up to orbit equivalence and von Neumann equivalence respectively.

math.DS↗
arXiv2009

Nonabelian free group actions: Markov processes, the Abramov-Rohlin formula and Yuzvinskii's formula

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 matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov-Rohlin formula for skew-product actions and Yuzvinskii's addition formula for algebraic actions.

math.DS↗
arXiv2008

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.

math.DS↗
arXiv2008

Free Groups in Lattices

Let G be any locally compact, unimodular, 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 perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If G/Γis noncompact then we require additional hypotheses that include G=SO(n,1).

math.GR↗
arXiv2008

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.

math.DS↗
arXiv2008

The Gromov Norm of the Product of Two Surfaces

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 of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.

math.GT↗
arXiv2006

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 $π_1(M)=Γ$ where $Γ$ is a lattice in $G=SO(n,1)$. The theorem can be rephrased in the following format. Let $X(\Z,Γ)$ be the space of representations of $\Z$ into $Γ$ modulo conjugation by $Γ$. $X(\Z,G)$ is defined similarly. Let $π: X(\Z,Γ)\to X(\Z,G)$ 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, $|π^{-1}(B_t)|$ 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.

math.GR↗
arXiv2005

A Solidification Phenomenon in Random Packings

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.

cond-mat.stat-mech↗
arXiv2005

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.

math.GT↗
arXiv2005

Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

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+ε) bi-Lipschitz homeomorphism between S'_1 and S'_2 and both covers S'_i have bounded injectivity radius. 2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j: S -> M where S is a surface of bounded injectivity radius and j is a pi_1-injective local isometry onto its image.

math.GT↗
arXiv2003

Periodicity and Circle Packing in the Hyperbolic Plane

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 isometry-invariant probability measures on the space of radius $r$-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius $r$-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.

math.MG↗
arXiv2003

Couplings of Uniform Spanniing Forests

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.

math.PR↗
arXiv2002

Optimally dense packings of hyperbolic space

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 ergodic theorem of Nevo to provide an appropriate definition of optimal dense packings. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.

math.MG↗