55. 
A brief introduction to
sofic entropy theory Sofic entropy theory is a generalization of the classical KolmogorovSinai entropy theory to actions of large class of nonamenable groups called sofic groups. This is a short introduction with a guide to the literature. 
submitted 
arXiv 

54. 
Finitary random interlacements and
the GaboriauLyons problem The von NeumannDay problem asks whether every nonamenable group contains a nonabelian free group. It was answered in the negative by Ol'shanskii in the 1980s. The measurable version (formulated by GaboriauLyons) asks whether every nonamenable measured equivalence relation contains a nonamenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a nonamenable group, extending work of GaboriauLyons. The proof uses an approximation to the random interlacement process by random multistep of geometricallykilled random walk paths. There are two applications: (1) the GaboriauLyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any nonamenable group, all Bernoulli shifts factor onto each other. 
submitted  arXiv 

53. 
All properly ergodic Markov chains
over a free group are orbit equivalent Previous work showed that all Bernoulli shifts over a free group are orbitequivalent. 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. 
submitted 
arXiv  
52. 
The space of stable weak
equivalence classes of measurepreserving actions The concept of (stable) weak
containment for measurepreserving 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 nonamenable 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 TuckerDrob  To appear in the Journal of Functional Analysis.  arXiv 
51. 
Examples in the entropy theory of
countable group actions KolmogorovSinai entropy is an invariant of measurepreserving 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) 
submitted 
arXiv 

50. 
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 
49. 
Zero entropy is generic Dan Rudolph showed that for an amenable group Γ, the generic measurepreserving 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 GaboriauLyons 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 

48. 
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 orbitequivalence between finitely generated groups extends to their end compactifications. 
Israel J. Math. 221 (2017), no. 1, 471–480. 
arXiv 

47. 
Hyperbolic geometry and pointwise
ergodic theorems We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of realrankone, 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  To appear in Ergodic Theory and Dynamical Systems.  arXiv 
46. 
von Neumann's problem and
extensions of nonamenable 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 nonamenable ergodic probability measure preserving (pmp) equivalence relation , the Bernoulli extension over a nonatomic 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 nontrivial probability space K. Second, we use this result to prove that any nonamenable 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 
To appear in Groups, Geometry and Dynamics.  arXiv 
45. 
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 finiteindex 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 

44. 
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. 
arXiv 

43. 
Mean convergence of Markovian
spherical averages for measurepreserving actions of
the free group Mean convergence of Markovian spherical averages is established for a measurepreserving action of a finitelygenerated 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 sigmaalgebra 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 
arXiv 
42. 
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 μentropy values of the ergodic, properly nonsingular Gactions 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. 
arXiv 
41. 
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 μstationary 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 μstationary, measurable Gactions 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 μstationary 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 GlasnerKing type 01 law stating that every dynamical property is either meager or residual. 
with Yair Hartman and Omer Tamuz  Trans. Amer. Math. Soc. 369 (2017), no. 7, 4889–4929.  arXiv 
40. 
Characteristic random subgroups of
geometric groups and free abelian groups of infinite
rank We show that if G is a nonelementary 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 Gergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary pgroups of countably infinite rank. 
with Rostislav Grigorchuk and Rostyslav Kravchenko  Trans. Amer. Math. Soc. 369 (2017), no. 2, 755–781.  arXiv 
39. 
L1measure equivalence and group
growth (Appendix to: Integrable measure equivalence for groups of polynomial growth by Tim Austin) We prove that group growth is an L1measure equivalence invariant. 
Groups Geom. Dyn. 10 (2016), no. 1, 117–154.  arXiv 

38. 
Weak density of orbit equivalence
classes of free group actions It is proven that the orbitequivalence class of any essentially free probabilitymeasurepreserving 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 

37. 
Cheeger constants and L2Betti
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 covolume. The existence depends on the L2Betti 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 4space and $\Gamma<\Isom(\H^4)$ is discrete and isomorphic to a subgroup of the fundamental group of a complete finitevolume hyperbolic 3manifold. Via PattersonSullivan 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 zeroth 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 PhillipsSarnak 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. 
arXiv 

36. 
vonNeumann 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 
arXiv 
35. 
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 nonamenable 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 
34. 
Entropy theory for sofic groupoids
I: the foundations This is the first part in a series in which sofic entropy theory is generalized to classbijective 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 nonfree Bernoulli shifts. The proofs are independent of previous literature. 
Journal d'Analyse Mathématique October 2014, Volume 124, Issue 1, pp 149233.  arXiv 

33. 
The type and stable type of the
boundary of a Gromov hyperbolic group Consider an ergodic nonsingular action $\Gamma \cc B$ of a countable group on a probability space. The type of this action codes the asymptotic range of the RadonNikodym derivative, also called the {\em ratio set}. If $\Gamma \cc X$ is a pmp (probabilitymeasurepreserving) 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 nonelementary Gromov hyperbolic group on its boundary with respect to a quasiconformal 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
363386. 
arXiv 

32. 
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 nonsingular actions of nonabelian free groups and establish a ratio ergodic theorem for averages along horospheres. 
with Amos Nevo 
Groups
Geom. Dyn. 8(2):331353, 2014.

arXiv 
31. 
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 CantorBendixson rank of $\Sub(G)$. The space of all conjugationinvariant 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 conjugationinvariant 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 
30. 
Invariant random
subgroups of the free group Let G be a locally compact group. A random closed subgroup with conjugationinvariant 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 StuckZimmer which show that there are no nonobvious IRS's of higher rank semisimple Lie groups with property (T). 
Groups
Geom. Dyn. 9 (2015), no. 3, 891–916. 
arXiv 

29. 
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 skewproduct setting. Using L. Bowen's finvariant we prove the addition theorem for actions of finitely generated free groups on skewproducts 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 
28. 
Harmonic models and
spanning forests of residually finite groups We prove a number of identities relating the sofic entropy of a certain class of nonexpansive 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), 17691808 
arXiv 
27. 
On a coinduction
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 $\CInd_H^G(aH) \times b_G$ 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 $\CInd_H^G(aH)$ weakly contains a whenever a is a Gaussian action. 
with Robin
TuckerDrob 
Israel
J. of Math. March 2013, Volume 194, Issue 1, pp 209224 
arXiv 
26. 
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 485510 
arXiv 

25. 
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, 777820 
arXiv 
24. 
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. 133164 
arXiv 
23. 
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 nontwoatom 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. 
arXiv 

22. 
Sofic entropy and
amenable groups In previous work, I introduced a measureconjugacy 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 measureconjugacy invariant called uppersofic entropy and a theorem of Rudolph and Weiss for the entropy of orbitequivalent actions relative to the orbit change σalgebra. 
Ergodic Theory Dynam. Systems 32 (2012), no. 2, 427–466  arXiv 

21. 
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 

20. 
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 orbitequivalent, 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 orbitequivalent. As an application, it is shown that if Γ is a free group, then all nontrivial Bernoulli shifts over Γ are orbitequivalent. 
Groups Geom. Dyn. 5 (2011), no. 1, 1–15.  arXiv 

19. 
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, 703718.  arXiv 

18. 
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, 93102 
arXiv 

17. 
Nonabelian
free
group actions: Markov processes, the AbramovRohlin
formula and Yuzvinskii's formula This paper introduces Markov chains and processes over nonabelian free groups and semigroups. We prove a formula for the finvariant 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 AbramovRohlin formula for skewproduct actions and Yuzvinskii's addition formula for algebraic actions. 
Ergodic
Theory
Dynam. Systems 30 (2010), no. 6, 16291663. 
arXiv 

Corrigendum We correct an error in the proof of the RohlinAbramov 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  
16. 
The
ergodic theory of free group actions: entropy and the
finvariant. Previous work introduced two measureconjugacy invariants: the finvariant (for actions of free groups) and Σentropy (for actions of sofic groups). The purpose of this paper is to show that the finvariant is a special case of Σentropy. There are two applications: the finvariant is invariant under group automorphisms and there is a uniform lower bound on the finvariant of a factor in terms of the original system. 
Groups Geom. Dyn. 4 (2010), no. 3, 419432  arXiv 

15. 
A
new measure conjugacy invariant for actions of free
groups This paper introduces a new measureconjugacy 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, 13871400.  arXiv 

14. 
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 measureconjugacy invariants for measurepreserving Gactions on probability spaces. These invariants generalize KolmogorovSinai 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 measureconjugacy 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), 217245.  arXiv 

13. 
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 BowenMarcus 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 HoweMoore 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, 97129.  arXiv 

12. 
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), 30213054.  arXiv 

11. 
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), 3767.  arXiv 

10. 
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, 10751089.  
9. 
Fluid/Solid
Transition
in a HardCore System We prove that a system of particles in the plane, interacting only with a certain hardcore constraint, undergoes a fluidsolid phase transition. 
with Russell Lyons, Charles Radin and Peter Winkler  Phys. Rev. Lett. 96, 025701 (2006)  
8. 
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 
7. 
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, 321339. erratum: Topology 47 (2008), no. 6, 471472.  arXiv 
6. 
Couplings
of
Uniform Spanning Forests We prove the existence of an automorphisminvariant 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, 21512158.  arXiv 

5. 
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), 3759.  arXiv 
4. 
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 ndimensional Euclidean or ndimensional hyperbolic space. We prove this conjecture. 
Geom. Dedicata 98 (2003), 211226.  arXiv 

3. 
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, 2339.  
2. 
Periodicity
and
Circle Packing in the Hyperbolic Plane We prove that given a fixed radius r, the set of isometryinvariant probability measures supported on ``periodic'' radius rcircle packings of the hyperbolic plane is dense in the space of all isometryinvariant probability measures on the space of radius rcircle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometryinvariant probability measures on a space of radius rpackings 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), 213236.  arXiv 

1. 
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). 