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Lewis Bowen

Publications and source records attributed to Lewis Bowen.

At least 55 records · Page 3Linked to original sources
arXiv2013

Cheeger constants and $L^2$-Betti numbers

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform lattice of $\Isom(X)$. As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form $\H^4/Γ$ where $\H^4$ is real hyperbolic 4-space and $Γ<\Isom(\H^4)$ is discrete and isomorphic to a subgroup of the fundamental group of a complete finite-volume hyperbolic 3-manifold. Via Patterson-Sullivan theory, this implies the existence of a uniform positive upper bound on the Hausdorff dimension of the conical limit set of such a $Γ$ when $Γ$ is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zero-th eigenvalue of the Laplacian of $\H^n/Γ$ over all discrete free groups $Γ<\Isom(\H^n)$ whenever $n\ge 4$ is even (the bound depends on $n$). This extends results of Phillips-Sarnak and Doyle who obtained such bounds for $n\ge 3$ when $Γ$ is a finitely generated Schottky group.

math.GT↗
arXiv2013

Amenable equivalence relations and the construction of ergodic averages for group actions

We present a general new method for constructing pointwise ergodic sequences on countable groups, which is applicable to amenable as well as to non-amenable groups and treats both cases on an equal footing. The principle underlying the method is that both cases can be viewed as instances of the general ergodic theory of amenable equivalence relations.

math.DS↗
arXiv2013

Entropy theory for sofic groupoids I: the foundations

This is the first part in a series in which sofic entropy theory is generalized to class-bijective extensions of sofic groupoids. Here we define topological and measure entropy and prove invariance. We also establish the variational principle, compute the entropy of Bernoulli shift actions and answer a question of Benjy Weiss pertaining to the isomorphism problem for non-free Bernoulli shifts. The proofs are independent of previous literature.

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arXiv2013

von-Neumann and Birkhoff ergodic theorems for negatively curved groups

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

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arXiv2012

The type and stable type of the boundary of a Gromov hyperbolic group

Consider an ergodic non-singular action $Γ\cc B$ 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 $Γ\cc X$ is a pmp (probability-measure-preserving) action, then the ratio set of the product action $Γ\cc B\times X$ is contained in the ratio set of $Γ\cc B$. So we define the {\em stable ratio set} of $Γ\cc B$ to be the intersection over all pmp actions $Γ\cc X$ of the ratio sets of $Γ\cc B\times X$. By analogy, there is a notion of {\em stable type} which codes the stable ratio set of $Γ\cc B$. 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 $III_λ$ for some $λ>0$. 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 $III_0$.

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arXiv2012

A Juzvinski\uı Addition Theorem for Finitely Generated Free Groups Actions

The classical Juzvinskiĭ Addition Theorem states that the entropy of an automorphism of a compact group decomposes along invariant subgroups. Thomas generalized the theorem to a skew-product setting. Using L. Bowen's f-invariant we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or compact Lie groups (correcting an error from [Bo10c]) and discuss examples.

math.DS↗
arXiv2012

A horospherical ratio ergodic theorem for actions of free groups

We prove a ratio ergodic theorem for amenable equivalence relations satisfying a strong form of the Besicovich covering property. We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio ergodic theorem for averages along horospheres.

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arXiv2011

Pointwise ergodic theorems beyond amenable groups

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this class contains all irreducible lattices in connected semisimple Lie groups without compact factors. We also establish similar results when the stable type is $III_λ$, $0 < λ< 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of p.m.p. actions of amenable groups to include p.m.p. amenable equivalence relations. Second, we show that it is possible to reduce the proof of ergodic theorems for p.m.p. actions of a general group to the proof of ergodic theorems in an associated p.m.p. amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

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arXiv2011

Harmonic models and spanning forests of residually finite groups

We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.

math.DS↗
arXiv2011

On a co-induction question of Kechris

This note answers a question of Kechris: if $H<G$ is a normal subgroup of a countable group $G$, $H$ has property MD and $G/H$ is amenable and residually finite then $G$ also has property MD. Under the same hypothesis we prove that for any action $a$ of $G$, if $b$ is a free action of $G/H$, and $b_G$ is the induced action of $G$ then $\CInd_H^G(a|H) \times b_G$ weakly contains $a$. Moreover, if $H<G$ is any subgroup of a countable group $G$, and the action of $G$ on $G/H$ is amenable, then $\CInd_H^G(a|H)$ weakly contains $a$ whenever $a$ is a Gaussian action.

math.GR↗
arXiv2011

Every countably infinite group is almost Ornstein

We say that a countable discrete group $G$ is {\em almost Ornstein} if for every pair of standard non-two-atom probability spaces $(K,κ), (L,λ)$ with the same Shannon entropy, the Bernoulli shifts $G \cc (K^G,κ^G)$ and $G \cc (L^G,λ^G)$ are isomorphic.

math.DS↗
arXiv2011

Sofic entropy and amenable groups

In previous work, I introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit change $σ$-algebra.

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arXiv2010

Geometric covering arguments and ergodic theorems for free groups

We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups. We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques. Some applications of our approach to other groups and other problems in ergodic theory are also briefly discussed.

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arXiv2010

Orbit equivalence, coinduced actions and free products

The following result is proven. Let $G_1 \cc^{T_1} (X_1,μ_1)$ and $G_2 \cc^{T_2} (X_2,μ_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let $Γ_i = G_i *H$ be the free product. Then the actions of $Γ_1$ and $Γ_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $Γ$ is a free group, then all nontrivial Bernoulli shifts over $Γ$ are orbit-equivalent.

math.DS↗
arXiv2010

An Isometry Between Measure Homology and Singular Homology

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 special case of hyperbolic manifolds as a consequence of his proof that simplicial volume is proportional to volume. We give a proof for all differentiable manifolds. This version corrects a few typos in an earlier version and formally proves the theorem for differentiable manifolds rather than locally finite simplicial complexes (but very few actual changes have been made).

math.GT↗