Random quotients of hyperbolic groups and Property (T)

Mr. Prayagdeep Parija
University of Wisconsin-Milwaukee
Dissertator

How does a random quotient of a group look like? Gromov looked at the
density model of quotients of free groups. The density parameter d measures
the proportion of the Cayley ball picked as relators. Using this
model, he proved that for d < 1/2, a typical quotient of a free group is
non-elementary hyperbolic. Ollivier extended Gromov’s result to show that
for d < 1/2 a typical quotient of even a non-elementary hyperbolic group is
non-elementary hyperbolic.
Zuk/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, 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).

Advisor: Prof. Chris Hruska

Committee Members:
Profs. Craig Guilbault, Jonah Gaster, Boris Okun, and Richard Stockbridge