14 citations · 47 across the 23 of their papers we have counts for
27 papers
The Structure of Almost Stationary Measures
Ilya Gekhtman, Simon Machado, Omri Solan +1
Let be a higher-rank simple Lie group acting on a space . A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on is either -invaria…
Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem
Ilya Gekhtman, Simon Machado, Omri Solan +1
Fraczyk and Gelander proved in \cite{FG} that for any simple Lie group of high rank and for every non-lattice discrete subgroup , the injectivity radius of points in $…
Topology and dynamics of unimodular random hyperbolic manifolds
Ilya Gekhtman, Nir Lazarovich, Arie Levit +1
We investigate the relationship between the space of ends of a unimodular random hyperbolic manifold and the dynamics of its geodesic flow. We show that having two ends of infinite…
Space-time boundaries for random walks and their application to operator algebras
Adam Dor-On, Ilya Gekhtman, Pavel Prudnikov
We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk with spectral radius and relate it to several classica…
Confined subgroups in groups with contracting elements
Inhyeok Choi, Ilya Gekhtman, Wenyuan Yang +1
In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is s…
Dynamics of subgroups of mapping class groups
Ilya Gekhtman, Biao Ma
Let be the mapping class group of a closed orientable surface of genus . Let be a non-elementary subgroup of so that the associated Bo…
Ratio-limit boundaries for random walks on relatively hyperbolic groups
Adam Dor-On, Matthieu Dussaule, Ilya Gekhtman
We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We extend, as well as determine significant limitati…
Stationary random subgroups in negative curvature
Ilya Gekhtman, Arie Levit
We show that discrete stationary random subgroups of isometry groups of Gromov hyperbolic spaces have full limit sets as well as critical exponents bounded from below. This informa…
Equidistribution of hyperbolic groups in homogeneous spaces
Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo
We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres…
Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichmüller space
Ilya Gekhtman, Yulan Qing, Kasra Rafi
We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the wo…
An embedding of the Morse boundary in the Martin boundary
Matthew Cordes, Matthieu Dussaule, Ilya Gekhtman
We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities g…
Central limit theorems for counting measures in coarse negative curvature
Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo
We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remov…
Stability phenomena for Martin boundaries of relatively hyperbolic groups
Matthieu Dussaule, Ilya Gekhtman
Let be a relatively hyperbolic group and let be an admissible symmetric finitely supported probability measure on . We extend Floyd-Ancona type inequalities up to the sp…
Entropy and drift for Gibbs measures on geometrically finite manifolds
Ilya Gekhtman, Giulio Tiozzo
We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) me…
Entropy and drift for word metric on relatively hyperbolic groups
Matthieu Dussaule, Ilya Gekhtman
We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random wa…
Counting conjugacy classes in groups with contracting elements
Ilya Gekhtman, Wen-yuan Yang
In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denot…
A central limit theorem for random closed geodesics: proof of the Chas-Li-Maskit conjecture
Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo
We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a…
Critical exponents of invariant random subgroups in negative curvature
Ilya Gekhtman, Arie Levit
Let be a proper geodesic Gromov hyperbolic metric space and let be a cocompact group of isometries of admitting a uniform lattice. Let be the Hausdorff dimension of…
Equidistribution of closed geodesics along random walk trajectories with respect to the harmonic invariant measure
Ilya Gekhtman
We prove that for suitable random walks on isometry groups of spaces, typical sample paths eventually land on loxodromic elements which equidistribute with respect to a f…
Counting problems in graph products and relatively hyperbolic groups
Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo
We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. th…
The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups
Matthieu Dussaule, Ilya Gekhtman, Victor Gerasimov +1
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give…
Martin boundary covers Floyd boundary
Ilya Gekhtman, Victor Gerasimov, Leonid Potyagailo +1
For finitely supported random walks on finitely generated groups we prove that the identity map on extends to a continuous equivariant surjection from the Martin boundary t…
The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds
Hyungryul Baik, Ilya Gekhtman, Ursula Hamenstaedt
We study the smallest positive eigenvalue of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold which fibers over the circle, with fiber a closed surface…
Exponential Torsion Growth for Random 3-Manifolds
Hyungryul Baik, David Bauer, Ilya Gekhtman +5
We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth.