Title：Invariant and stationary random subgroups and their applications in geometry
Speaker: Ilya Gekhtman (Technion – Israel Institute of Technology)
Abstract: Invariant random subgroups are conjugation invariant measures on the space of subgroups of a given group G. Their study provides a fertile playground for the interaction of algebra, geometry, probability and ergodic theory. On one hand, they arise as point stabilizers for probability measure preserving actions. On the other hand, they provide a vast generalization of both normal subgroups of countable groups and lattices in Lie groups. On the third hand, they (and their relatives, the so-called stationary random subgroups) have applications ranging from proving Margulis's celebrated normal subgroup theorem (which asserts that any normal subgroup of a lattice in a higher rank simple Lie group is finite index) to compactifying the moduli space of Riemann surfaces, to studying the injectivity radius of hyperbolic manifolds. In this minicourse, we will give an overview of invariant and stationary random subgroups and their applications. Special attention will be given to invariant random subgroups of semisimple Lie groups and of hyperbolic groups.
Time and Venue:
03/12 （Wednesday): 13:30-15:30, Zhizhi Building 12
03/13 (Thursday): 13:30-15:30, Zhizhi Building 12
03/14 (Friday): 13:30-15:30, Zhizhi Building 12
About the speaker: Ilya Gekhtman is an assistant professor of mathematics at the Technion in Haifa, Israel. He earned his Ph.D. from the University of Chicago in 2014 and subsequently held prestigious postdoctoral positions at Yale University, the University of Bonn, and the University of Toronto. His research lies at the intersection of ergodic theory, probability, geometric group theory, and hyperbolic geometry, with a particular focus on random walks on hyperbolic groups, invariant and stationary random subgroups, Teichmüller dynamics, and asymptotic counting problems in group theory and geometry. He has published in leading mathematical journals, including Inventiones Mathematicae, GAFA, Compositio Mathematica, Journal of Topology, Advances in Mathematics, IMRN, and Probability Theory and Related Fields.
Lecture 1: Introduction to invariant random subgroups: definitions, examples, basic properties and applications.
Lecture 2: Random walks on groups and stationary random subgroups. Growth rates of invariant and stationary random subgroups of Lie groups and hyperbolic groups.
Lecture 3: Applications of stationary random subgroups to injectivity radius of hyperbolic manifolds and other locally symmetric spaces.
