Research Papers

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Finitary random interlacements and the Gaboriau-Lyons problem

The von Neumann-Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol'shanskii in the 1980s. The measurable version (formulated by Gaboriau-Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau-Lyons. The proof uses an approximation to the random interlacement process by random multistep of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau-Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.

All properly ergodic Markov chains over a free group are orbit equivalent

Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group. A list of related open problems is provided.

The space of stable weak equivalence classes of measure-preserving actions

The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when Γ = Z this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.
with Robin Tucker-Drob submitted
Examples in the entropy theory of countable group actions

Kolmogorov-Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

(this is a 130+ page survey article with many new examples)

Invariant random subgroups of semidirect products

We study invariant random subgroups (IRSs) of semidirect products G=A⋊Γ. In particular, we characterize all IRSs of parabolic subgroups of SLd(ℝ), and show that all ergodic IRSs of ℝ^d⋊SLd(ℝ) are either of the form ℝ^d⋊K for some IRS of SLd(ℝ), or are induced from IRSs of Λ⋊SL(Λ), where Λ<ℝ^d is a lattice.
with Ian Biringer and Omer Tamuz
submitted arXix
Zero entropy is generic

Dan Rudolph showed that for an amenable group Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other.

Entropy 18 (2016), no. 6, Paper No. 220, 20 pp. arXiv
Integrable orbit equivalence rigidity for free groups

It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.

to appear in Israel J. of Math. arXiv
Hyperbolic geometry and pointwise ergodic theorems

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.
with Amos Nevo submitted
von Neumann's problem and extensions of non-amenable equivalence relations

The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons \cite{GL07} to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation , the Bernoulli extension over a non-atomic base space (K,κ) contains the orbit equivalence relation of a free ergodic pmp action of 𝔽2. Moreover, we provide conditions which imply that this holds for any non-trivial probability space K. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).
with Daniel Hoff and Adrian Ioana
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.

Proc. Amer. Math. Soc. 145 (2017), no. 1, 215–224 arXiv
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.

to appear in Ergodic Theory and Dynamical Systems
Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix Π defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.
with Alexander Bufetov and Olga Romaskevich
Geom. Dedicata 181 (2016), 293–306
Property (T) and the Furstenberg entropy of nonsingular actions

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition.
with Yair Hartman and Omer Tamuz Proc. Amer. Math. Soc. 144 (2016), no. 1, 31–39.
Generic stationary measures and actions

Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space.

Equip the space of stationary measures with the weak* topology. When μ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G,μ). When Z is compact, this implies that the simplex of μ-stationary measures on ZG is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0,1}G.
We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary.

Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property (T), the ergodic actions are meager. We also construct a group G without property (T) such that the ergodic actions are not dense, for some μ.
Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.

with Yair Hartman and Omer Tamuz to appear in Transactions of the AMS arXiv
Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

We show that if G is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then G has 2ℵ0 many continuous ergodic invariant random subgroups. If G is a nonabelian free group then G has 2ℵ0 many continuous G-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary p-groups of countably infinite rank.
with Rostislav Grigorchuk and Rostyslav Kravchenko Trans. Amer. Math. Soc. 369 (2017), no. 2, 755–781. arXiv
L1-measure equivalence and group growth
(Appendix to: Integrable measure equivalence for groups of polynomial growth by Tim Austin)

We prove that group growth is an L1-measure equivalence invariant.

Groups Geom. Dyn. 10 (2016), no. 1, 117–154. arXiv
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.

Groups Geom. Dyn. 9 (2015), no. 3, 811–830. arXiv
Cheeger constants and L2-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 $\Gamma<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the L2-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 $\H^4/\Gamma$ where $\H^4$ is real hyperbolic 4-space and $\Gamma<\Isom(\H^4)$ 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 $\H^n/\Gamma$ over all discrete free groups $\Gamma<\Isom(\H^n)$ whenever n≥4 is even (the bound depends on n). This extends results of Phillips-Sarnak and Doyle who obtained such bounds for n≥3 when Γ is a finitely generated Schottky group.

Duke Math. J. 164 (2015), no. 3, 569–615.
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.
with Amos Nevo Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 5, 1113–1147
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.
with Amos Nevo J. Anal. Math. 126 (2015), 359–388. arXiv
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.

Journal d'Analyse Mathématique , Volume 124, Issue 1, pp 149--233. arXiv
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 IIIλ 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 III0 and, if it is weakly mixing, then it is not stable type III0.

Geometriae Dedicata, October 2014, Volume 172, Issue 1, pp 363--386.
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.
with Amos Nevo Groups Geom. Dyn. 8(2):331--353, 2014. arXiv
Invariant random subgroups of lamplighter groups (new version as of Sept 2, 2013)

Let G be one of the lamplighter groups $({\mathbb{Z}/p\bz})^n\wr\mathbb{Z}$ and $\Sub(G)$ the space of all subgroups of G. We determine the perfect kernel and Cantor-Bendixson rank of $\Sub(G)$. The space of all conjugation-invariant Borel probability measures on $\Sub(G)$ is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If F is a finite group and Γ an infinite group which does not have property (T) then the conjugation-invariant probability measures on $\Sub(F\wr\Gamma)$ supported on ⊕ΓF also form a Poulsen simplex.
with Rostislav Grigorchuk and Rostyslav Kravchenko
Israel J. Math. 207 (2015), no. 2, 763–782. arXiv
Invariant random subgroups of the free group

Let G be a locally compact group. A random closed subgroup with conjugation-invariant law is called an {\em invariant random subgroup} or IRS for short. We show that each nonabelian free group has a large "zoo" of IRS's. This contrasts with results of Stuck-Zimmer which show that there are no non-obvious IRS's of higher rank semisimple Lie groups with property (T).

Groups Geom. Dyn. 9 (2015), no. 3, 891–916.
A Juzvinskiĭ Addition Theorem for Finitely Generated Free Group Actions

The classical Juzvinski\u{i} 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.

with Yonatan Gutman
Ergodic Theory Dynam. Systems 34 (2014), no. 1, 95–109. arXiv
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.
with Hanfeng Li
J. Funct. Anal. 263, no. 7, (2012), 1769--1808
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 bG 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.

with Robin Tucker-Drob
Israel J. of Math. March 2013, Volume 194, Issue 1, pp 209--224
Random walks on 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.

Invent. Math.  Volume 196, Issue 2 (2014), Page 485-510
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 III1. 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 IIIλ, 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.

with Amos Nevo Ergod. Th. and Dynam. Sys. (2013), 33, 777--820

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.

with Amos Nevo L’Enseignement Mathématique, Volume 59, Issue 1/2, 2013, pp. 133--164

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 $G \cc (K^G,\kappa^G)$ and $G \cc (L^G,\lambda^G)$ are isomorphic.

Dynamical systems and group actions, 67–78, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012.
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.

Ergodic Theory Dynam. Systems 32 (2012), no. 2, 427–466 arXiv
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 G1,G2 are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over G1 is stably orbit equivalent to every nontrivial Bernoulli shift over G2. This answers a question of S. Popa.

Groups Geom. Dyn. 5 (2011), no. 1, 17–38. arXiv
Orbit equivalence, coinduced actions and free products

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 G1 and G2. Let H be any countable group. For i=1,2, let Γi=Gi∗H be the free product. Then the actions of Γ1 and Γ2 coinduced from T1 and T2 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.

Groups Geom. Dyn. 5 (2011), no. 1, 1–15. arXiv
Entropy for expansive algebraic actions of residually finite groups

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

Ergodic Theory Dynam. Systems. 31 (2011), no. 3, 703--718. arXiv
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.

Israel J. of Math, (2011) Volume 183, Number 1, 93-102
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.

Ergodic Theory Dynam. Systems 30 (2010), no. 6, 1629--1663.


We correct an error in the proof of the Rohlin-Abramov addition formula for free group actions and point out errors in the proof of Yuzvinskii's addition formula. It is not known if the latter are fixable.

with Yonatan Gutman Ergodic Theory and Dynam. Systems  33 Issue 02  / April 2013, pp 643 - 645 arXiv
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.

Groups Geom. Dyn. 4 (2010), no. 3, 419--432 arXiv
A new measure conjugacy invariant for actions of free groups

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.

Ann. of Math., vol. 171 (2010), No. 2, 1387--1400. arXiv
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.

J. Amer. Math. Soc., 23 (2010), 217-245. arXiv
Invariant measures on the space of horofunctions of a word hyperbolic group

We introduce a natural equivalence relation on the space $\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if η is such a measure and G acts on a space (X,μ) by p.m.p. transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

Ergodic Theory Dynam. Systems. 30 (2010), no. 1, 97--129. arXiv
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 \Gamma < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of \Gamma. If G/\Gamma is noncompact then we require additional hypotheses that include G=SO(n,1).

Geometry & Topology, 13, (2009), 3021--3054. arXiv
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 π1(M)=Γ where Γ 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 Γ modulo conjugation by Γ. $X(\Z,G)$ 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 {Bt}, $B_t \subset X(\Z,G)$ satisfying certain explicit hypotheses, |π−1(Bt)| is asymptotic to vol(Bt).
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.

Geom. Dedicata, 124, (2007), 37--67. arXiv
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.
with Russell Lyons, Charles Radin and Peter Winkler SIAM J. Math. Anal. 38 (2006), no. 4, 1075--1089.
Fluid/Solid Transition in a Hard-Core System

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid-solid phase transition.
with Russell Lyons, Charles Radin and Peter Winkler Phys. Rev. Lett. 96, 025701 (2006)
Uniqueness and symmetry in problems of optimally dense packings

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 of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hyperbolic plane. Our goal is to understand qualitative features of such optimum density problems, in particular the appropriate meaning of the uniqueness of solutions, and the role of symmetry in classfying optimally dense packings.

with Charles Holton, Charles Radin and Lorenzo Sadun Math. Phys. Electron. J. 11 (2005), Paper 1, 34 pp. arXiv
On 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.
with Mike Develin, Jesus De Loera and Francisco Santos Topology 44 (2005), no. 2, 321--339. erratum: Topology 47 (2008), no. 6, 471--472. arXiv
Couplings of Uniform Spanning 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.

Proc. Amer. Math. Soc. 132 (2004), no. 7, 2151--2158. arXiv
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.

with Charles Radin Geom. Dedicata 104 (2004), 37--59. arXiv
On the existence of completely saturated packings and completely reduced coverings

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.

Geom. Dedicata 98 (2003), 211--226. arXiv
Densest Packing of Equal Spheres in Hyperbolic Space

We consider circle packings in the hyperbolic plane, by finitely many congru- ent circles, which maximize the number of touching pairs. We show that such a packing has all of its centers located on the vertices of a triangulation of the hyperbolic plane by congruent equilateral triangles, provided the diameter d of the circles is such that an equilateral triangle in the hyperbolic plane of side length d has each of its angles is equal to 2π/N for some N > 6.
with Charles Radin
Discrete Comput. Geom. 29 (2003), no. 1, 23--39. 
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.

Geom. Dedicata 102 (2003), 213--236. arXiv
Circle Packing in Hyperbolic Space

We consider circle packings in the hyperbolic plane, by finitely many congru- ent circles, which maximize the number of touching pairs. We show that such a packing has all of its centers located on the vertices of a triangulation of the hyperbolic plane by congruent equilateral triangles, provided the diameter d of the circles is such that an equilateral triangle in the hyperbolic plane of side length d has each of its angles is equal to 2π/N for some N > 6.

Math. Phys. Electron. J. 6 (2000), Paper 6, 10 pp. (electronic).

This material is based upon work supported by the National Science Foundation under Grants No. 0901835 (9/1/09-2/28/10), 0968762 (9/1/09-8/31/12) and 0954606 (5/15/10-4/30/15). The collaborative work with Amos Nevo was supported by the Binational Science Foundation under Grant No. 2008274 (9/1/09-8/31/13).

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF) or the Binational Science Foundation.