# Research Papers

 62. Locally compact sofic groups We introduce the notion of soficity for locally compact groups and list a number of open problems. with Peter Burton To appear in Israel J. of Math. arXiv 61. A multiplicative ergodic theorem for von Neumann algebra valued cocycles The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution. with Ben Hayes, Yuqing (Frank) Lin Comm. Math. Phys. 384 (2021), no. 2, 1291–1350. arXiv 60. A topological dynamical system with two different positive sofic entropies A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs. with Dylan Airey and Yuqing (Frank) Lin to appear in Trans. Amer. Math. Soc. arXiv video talk slides 59. Flexible stability and nonsoficity A sofic group G is said to be flexibly stable if every sofic approximation to G can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if PSLd(ℤ) is flexibly stable for some d≥5 then there exists a group which is not sofic. with Peter Burton arXiv 58. Failure of the L^1 pointwise ergodic theorem for PSL(2,R). Amos Nevo established the pointwise ergodic theorem in Lp for measure-preserving actions of PSL2(ℝ) on probability spaces with respect to ball averages and every p>1. This paper shows by explicit example that Nevo's Theorem cannot be extended to p=1. with Peter Burton Geom. Dedicata 207 (2020), 61–80. arXiv 57. Sofic homology invariants and the weak Pinsker property A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every  ε>0 it is measurably conjugate to the direct product of a transformation with entropy <ε and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group Γ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a Markov chain over the free group which arises as a weak* limit of the hardcore model on random regular graphs. In order to prove that it does not have the WPP, we introduce a large family of new measure conjugacy invariants based on the homology growth of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential 0-dimensional homology, while the counterexample has a positive exponential growth rate of 0-dimensional homology. to appear in Amer. J. Math. arXiv 56. Superrigidity, measure equivalence, and weak Pinsker entropy We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G. to appear in Groups, Geometry and Dynamics. arXiv 55. A brief introduction to sofic entropy theory Sofic entropy theory is a generalization of the classical Kolmogorov-Sinai entropy theory to actions of large class of non-amenable groups called sofic groups. This is a short introduction with a guide to the literature. Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, 1847–1866, World Sci. Publ., Hackensack, NJ, 2018 arXiv video talk slides 54. 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. Geom. Funct. Anal. 29 (2019), no. 3, 659–689. 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 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. Unimodularity in randomly generated graphs, 155–174, Contemp. Math., 719, Amer. Math. Soc., Providence, RI, 2018. arXiv 52. 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 J. Funct. Anal. 274 (2018), no. 11, 3170–3196. arXiv 51. 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) Ergodic Theory Dynam. Systems 40 (2020), no. 10, 2593–2680. 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 Ergodic Theory Dynam. Systems 40 (2020), no. 2, 353–366. arXix 49. 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 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 orbit-equivalence 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 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 Ergodic Theory Dynam. Systems 39 (2019), no. 10, 2689–2716 arXiv 46. 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 Groups Geom. Dyn. 12 (2018), no. 2, 399–448. 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 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 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. Ergodic Theory Dynam. Systems 38 (2018), no. 7, 2447–2492. arXiv 43. 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 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 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. 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 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 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 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 39. 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 38. 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 37. 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. arXiv 36. 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 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 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 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 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 October 2014, Volume 124, Issue 1, pp 149--233. arXiv 33. 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. 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 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 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 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 30. 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. 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 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 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 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 arXiv 27. On a co-induction question of Kechris This note answers a question of Kechris: if H 6. with Charles Radin Discrete Comput. Geom. 29 (2003), no. 1, 23--39. 2. 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 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).