Date of Award

August 2023

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Chris Hruska

Committee Members

Craig Guilbault, Richard Stockbridge, Boris Okun, Jonah Gaster


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.

Żuk 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

Included in

Mathematics Commons