13 results on '"Ilya Gekhtman"'
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2. Entropy and drift for word metrics on relatively hyperbolic groups
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Ilya Gekhtman and Matthieu Dussaule
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Pure mathematics ,Rank (linear algebra) ,010102 general mathematics ,16. Peace & justice ,Random walk ,01 natural sciences ,Moment (mathematics) ,Entropy (classical thermodynamics) ,0103 physical sciences ,Discrete Mathematics and Combinatorics ,010307 mathematical physics ,Geometry and Topology ,Finitely-generated abelian group ,0101 mathematics ,Abelian group ,Word (group theory) ,Mathematics ,Word metric - Abstract
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 walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blachere, Haissinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hy-perbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict. more...
- Published
- 2020
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3. Counting problems in graph products and relatively hyperbolic groups
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Samuel J. Taylor, Giulio Tiozzo, and Ilya Gekhtman
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Large class ,Full density ,Cayley graph ,General Mathematics ,010102 general mathematics ,Coxeter group ,Geometric Topology (math.GT) ,Dynamical Systems (math.DS) ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,Mathematics - Geometric Topology ,Counting problem ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Group Theory ,Word metric ,Mathematics - Abstract
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., they have full density with respect to counting in balls for the word metric in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups. more...
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- 2020
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4. Entropy and drift for Gibbs measures on geometrically finite manifolds
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Giulio Tiozzo and Ilya Gekhtman
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Entropy (statistical thermodynamics) ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Geometric Topology (math.GT) ,Dynamical Systems (math.DS) ,Random walk ,01 natural sciences ,Mathematics - Geometric Topology ,Metric space ,Corollary ,FOS: Mathematics ,Mathematics - Dynamical Systems ,60G50, 37D35, 53D25, 60J50 ,0101 mathematics ,Critical exponent ,Mathematics - Probability ,Quotient ,Mathematics - Abstract
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) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density., Comment: 29 pages more...
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- 2020
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5. An embedding of the Morse boundary in the Martin boundary
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Matthew Cordes, Matthieu Dussaule, Ilya Gekhtman, Department of Mathematics [ETH Zurich] (D-MATH), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Department of Mathematics [University of Toronto], University of Toronto, Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology in Zürich [Zürich] (ETH Zürich), Université de Nantes - Faculté des Sciences et des Techniques, and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS) more...
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[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Mathematics - Metric Geometry ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,Probability (math.PR) ,FOS: Mathematics ,Metric Geometry (math.MG) ,Geometry and Topology ,Group Theory (math.GR) ,[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] ,Mathematics - Group Theory ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Mathematics - Probability - Abstract
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 generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic. more...
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- 2020
6. Exponential Torsion Growth for Random 3-Manifolds
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Hyungryul Baik, Thorben Kastenholz, Ilya Gekhtman, David Bauer, Sebastian Hensel, Daniel Valenzuela, Bram Petri, and Ursula Hamenstädt
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Pure mathematics ,Betti number ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,0102 computer and information sciences ,Homology (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,57M10, 57Q10 ,Exponential function ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,010201 computation theory & mathematics ,FOS: Mathematics ,Torsion (algebra) ,0101 mathematics ,Mathematics - Abstract
We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth., Comment: 32 pages, no figures
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- 2017
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7. The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups
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Leonid Potyagailo, Ilya Gekhtman, Matthieu Dussaule, Victor Gerasimov, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS) more...
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Pure mathematics ,Group (mathematics) ,Hyperbolic space ,Probability (math.PR) ,Geometric topology (object) ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Boundary (topology) ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Dynamical Systems (math.DS) ,Random walk ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Mathematics - Geometric Topology ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,FOS: Mathematics ,Finitely generated group ,Mathematics - Dynamical Systems ,Abelian group ,Mathematics - Group Theory ,Mathematics - Probability ,Group theory ,Mathematics - Abstract
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 a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space ${\mathcal H}^n$, we show that the Martin boundary coincides with the $CAT(0)$ boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet., Oberwolfach Preprints;2018,03 more...
- Published
- 2020
8. Stability phenomena for Martin boundaries of relatively hyperbolic groups
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Ilya Gekhtman, Matthieu Dussaule, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN) more...
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Statistics and Probability ,Pure mathematics ,Spectral radius ,Boundary (topology) ,Group Theory (math.GR) ,Type (model theory) ,01 natural sciences ,Relatively hyperbolic group ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Mathematics - Geometric Topology ,010104 statistics & probability ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Probability measure ,Kleinian group ,Probability (math.PR) ,010102 general mathematics ,Geometric Topology (math.GT) ,Homeomorphism ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Statistics, Probability and Uncertainty ,Degeneracy (mathematics) ,Mathematics - Group Theory ,Mathematics - Probability ,Analysis - Abstract
Let $$\Gamma $$ be a relatively hyperbolic group and let $$\mu $$ be an admissible symmetric finitely supported probability measure on $$\Gamma $$ . We extend Floyd–Ancona type inequalities from Gekhtman et al. (Martin boundary covers Floyd boundary, 2017. arXiv:1708.02133 ) up to the spectral radius R of $$\mu $$ . We use them to find the precise homeomorphism type of the r-Martin boundary, which describes r-harmonic functions, for every $$r\le R$$ . We also define a notion of spectral degeneracy along parabolic subgroups which is crucial to describe the homeomorphism type of the R-Martin boundary. Finally, we give a criterion for (strong) stability of the Martin boundary in the sense of Picardello and Woess (in: Potential theory, de Gruyter, 1992) in terms of spectral degeneracy. We then prove that this criterion is always satisfied in small rank, so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable. more...
- Published
- 2019
9. Critical exponents of invariant random subgroups in negative curvature
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Ilya Gekhtman and Arie Levit
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Group Theory (math.GR) ,Dynamical Systems (math.DS) ,01 natural sciences ,Combinatorics ,Mathematics - Geometric Topology ,0103 physical sciences ,FOS: Mathematics ,Ergodic theory ,Mathematics::Metric Geometry ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Dynamical Systems ,Mathematics ,Maximal ergodic theorem ,Gromov boundary ,Simple Lie group ,010102 general mathematics ,Geometric Topology (math.GT) ,Locally compact group ,Metric space ,Hausdorff dimension ,010307 mathematical physics ,Geometry and Topology ,22F10, 20F67, 20F69, 20H10, 20H10, 22E40, 22D40 ,Mathematics - Group Theory ,Analysis - Abstract
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical exponent $\delta(\mu)$ of any discrete invariant random subgroup $\mu$ of the locally compact group $G$ and show that $\delta(\mu) > \frac{d}{2}$ in general and that $\delta(\mu) = d$ if $\mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdan's property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo., Comment: Accepted to be published in GAFA more...
- Published
- 2018
10. Martin boundary covers Floyd boundary
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Ilya Gekhtman, Wenyuan Yang, Victor Gerasimov, and Leonid Potyagailo
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Pure mathematics ,General Mathematics ,Mathematics::General Topology ,Boundary (topology) ,Group Theory (math.GR) ,Dynamical Systems (math.DS) ,01 natural sciences ,Surjective function ,Mathematics - Geometric Topology ,0103 physical sciences ,FOS: Mathematics ,Identity function ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics ,Probability (math.PR) ,010102 general mathematics ,Geometric topology (object) ,Geometric Topology (math.GT) ,Random walk ,20F65, 20P05, 37D40 ,Equivariant map ,010307 mathematical physics ,Finitely generated group ,Mathematics - Group Theory ,Group theory ,Mathematics - Probability - Abstract
For finitely supported random walks on finitely generated groups $G$ we prove that the identity map on $G$ extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key estimate relates the Green and Floyd metrics, generalizing results of Ancona for random walks on hyperbolic groups and of Karlsson for quasigeodesics. We then apply these techniques to obtain some results concerning the harmonic measure on the limit sets of geometrically finite isometry groups of Gromov hyperbolic spaces. . more...
- Published
- 2017
11. A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture
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Samuel J. Taylor, Ilya Gekhtman, and Giulio Tiozzo
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Pure mathematics ,Conjecture ,Geodesic ,General Mathematics ,010102 general mathematics ,Mathematics::General Topology ,Surface (topology) ,Mathematics::Geometric Topology ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Special case ,Pair of pants ,Central limit theorem ,Mathematics - Abstract
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 conjecture of Chas–Li–Maskit. more...
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- 2019
- Full Text
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12. The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds
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Ursula Hamenstädt, Hyungryul Baik, and Ilya Gekhtman
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Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Geometric Topology ,Geodesic ,Differential Geometry (math.DG) ,General Mathematics ,FOS: Mathematics ,58C40, 30F60, 20P05 ,Fibered knot ,Geometric Topology (math.GT) ,Mathematics::Spectral Theory ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We study the smallest positive eigenvalue $\lambda_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $\lambda_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $\lambda_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements. more...
- Published
- 2016
13. Counting loxodromics for hyperbolic actions
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Giulio Tiozzo, Ilya Gekhtman, and Samuel J. Taylor
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Pure mathematics ,Geodesic ,Gromov boundary ,Hyperbolic group ,010102 general mathematics ,Structure (category theory) ,Geometric Topology (math.GT) ,Dynamical Systems (math.DS) ,Group Theory (math.GR) ,01 natural sciences ,Measure (mathematics) ,Mathematics - Geometric Topology ,Metric space ,0103 physical sciences ,FOS: Mathematics ,Ergodic theory ,010307 mathematical physics ,Geometry and Topology ,Ball (mathematics) ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the Gromov boundary $\partial X$. Our techniques make use of the automatic structure of $G$, Patterson--Sullivan measure on $\partial G$, and the ergodic theory of random walks for groups acting on hyperbolic spaces. We discuss various applications, in particular to Mod(S), Out($F_N$), and right--angled Artin groups. more...
- Published
- 2016
- Full Text
- View/download PDF
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