Ilya Gekhtman

I am an assistant professor of mathematics at the Technion in Haifa, Israel.

I received my Ph.D from the University of Chicago in 2014 and subsequently held postdoctoral positions at the universities of Yale, Bonn, and Toronto.

My research interests are at the nexus of ergodic theory, probability, geometric group theory, and hyperbolic geometry.

Specifically I am interested in random walks on hyperbolic groups and their relatives, asymptotic counting problems in group theory and geometry, invariant and stationary random subgroups, Patterson-Sullivan theory, Teichmueller dynamics, and dynamics in negative curvature.

Email: ilyagekh@gmail.com

Research statement

Teaching statement

Publications:

I. Gekhtman, S. Taylor, and G. Tiozzo, "Central limit theorems for counting measures in coarse negative curvature ," Compositio Mathematica , Volume 158 , Issue 10 , October 2022 , pp. 1980 - 2013, arxiv:2004.13084

M. Cordes. M. Dussaule, and I. Gekhtman, "An embedding of the Morse boundary in the Martin boundary," Algebraic and Geometric Topology; 2022; Volume 22, 1217-1253. arXiv:2004.14624

I. Gekhtman and W. Yang, “Counting conjugacy classes in groups with contracting elements,” Journal of Topology, v15(2) 620-665 2022 arXiv:1810.02969

I. Gekhtman, V. Gerasimov, L. Potyagailo, and W. Yang, “Martin boundary covers Floyd boundary,” to appear, Inventiones Mathematicae, no 223, 759–809(2021) arXiv:1708.02133

M. Dussaule and I. Gekhtman, “Stability phenomena for Martin boundaries of relatively hyperbolic groups,” Probability Theory and Related Fields, 2020, Volume 179, 201-259 arXiv:1909.0157

M. Dussaule, I. Gekhtman, V. Gerasimov, and L. Potyagailo, “Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups,” L'Enseignement mathématique , 2021, Volume 66, 341-382. arXiv:1711.11307

I. Gekhtman and G. Tiozzo, “Entropy and drift for Gibbs measures on geometrically finite manifolds,” Transactions of the American Mathematical Society 373 (2020), 2949-2980 , arXiv:1904.01187

I. Gekhtman, S. Taylor, and G. Tiozzo, “A central limit theorem for random closed geodesics: proof of the Chas-Li-Maskit conjecture,” Advances in Mathematics 358 (2019) , arXiv:1808.08422

M. Dussaule and I. Gekhtman, “Entropy and drift for word metrics in relatively hyperbolic groups,” Groups, Geometry and Dynamics, 2020, Volume 14, 1455-1509. arXiv:1811.10849

I. Gekhtman, S. Taylor, and G. Tiozzo, “Counting loxodromics in graph products and relatively hyperbolic groups,” Israel Journal of Mathematics, 237 (2020), 311-371, arXiv:1711.04177

H. Baik, I. Gekhtman, and U. Hamenstaedt, “The Smallest positive eigenvalue of fibered hyperbolic 3-manifolds,” Proceedings of the London Mathematical Society(2019), Volume 120, Issue 5, 704-741, arXiv:1608.07609

I. Gekhtman and A. Levit, “Critical exponents for invariant random subgroups in negative curvature,” GAFA (2019) 24: 1411-1439, arXiv:1804.02995

I. Gekhtman, S. Taylor, and G. Tiozzo, “Counting loxodromics for hyperbolic actions,” J. Topol. 11 (2018), no. 2, 379-419, arXiv:1605.02103

H. Baik, D. Bauer, I. Gekhtman, U. Hamenstädt, S. Hensel, T. Kastenholz, B. Petri, D. Valenzuela, “Exponential torsion growth for random 3-manifolds,” Int. Math. Res. Not. IMRN 2018, no. 21, 6497-6534, arXiv:1607.00631

Preprints:

I. Gekhtman, K. Rafi and Y. Qing, Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichmüller space,” arXiv:arXiv:2208.04778 .

I. Gekhtman, S. Taylor, and G. Tiozzo, “Equidistribution of hyperbolic groups in homogeneous spaces," arxiv:arXiv:2209.06776

I. Gekhtman, “Equidistribution of closed geodesics along random walk trajectories with respect to the harmonic invariant measure,”arXiv:1711.04985

I. Gekhtman, “Stable type of the mapping class group,”arXiv:1310.5364

I. Gekhtman, “Dynamics of convex-cocompact subgroups of mapping class groups,” arxiv:1204.1741

Expository notes not for publication

I. Gekhtman and A. Wright, Smillie's Theorem on closed SL_2(R) orbits of quadratic differentials

Some slides from presentations

Equilibrium states and random walks in negative curvature

Invariant random subgroups in negative curvature and their growth rates

Martin boundaries of relatively hyperbolic groups

Personal

I was born in Kiev, Ukraine, but grew up in northern Indiana, where I graduated from Penn High School.

I then attended Caltech in Pasadena, California.

I am married to Galyna Dobrovolska.

I support this Statement of Inclusiveness.

I like long-distance running.