Time:May 21, 2025,10:30 (Beijing time)
Speaker: Prayagdeep Parija (Virginia Tech)
Title: Random quotients of hyperbolic groups and Property (T)
Abstract : What does a typical quotient of a group look like? Gromov looked at the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d < 1/2, the typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov’s result to show that for d < 1/2, the typical quotient of many hyperbolic groups is also non-elementary hyperbolic.
Zuk and Kotowski–Kotowski proved that for d > 1/3, a typical quotient of a free group has Property (T). We show that (in a closely related density model) for 1/3 < d < 1/2, the typical quotient of a large class of hyperbolic groups is non-elementary hyperbolic and has Property (T). This provides an answer to a question of Gromov (and Ollivier).
