Prayagdeep Parija: Random Quotients of Hyperbolic Groups and Property (T)

Prayagdeep Parija, University of Wisconsin Milwaukee
Title: Random Quotients of Hyperbolic Groups and Property (T)
What does a typical quotient of a group look like? Gromov had looked at 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 had proved that for $d$ less than $1/2$ a typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov's result to show that for $d$ less than $1/2$ a typical quotient of even a non-elementary hyperbolic group is non-elementary hyperbolic.

{\.Z}uk/Kotowski-Kotowski proved that for $d$ greater than $1/3$ a typical quotient of a free group has Property-(T).We show that (in a closely related density model) for $d$ between $1/3$ and $1/2$ a typical quotient of a non-elementary hyperbolic group is non-elementary hyperbolic and has Property-(T). This provides an answer to a question of Gromov (and Ollivier).