School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

University of Chicago

A famous question of Gromov asks whether every hyperbolic

Abstract:

group contains a subgroup which is isomorphic to the fundamental group of

a closed surface. Surface subgroups play a very important role in many

areas of low-dimensional topology, for example in Agol's recent proof that

every hyperbolic 3-manifold has a finite cover which fibers over the circle. I

would like to describe several ways to build surface subgroups in certain

hyperbolic groups. The role of hyperbolicity is twofold here: first,

hyperbolic geometry allows one to certify injectivity by *local* data;

second, hyperbolic dynamics allows one to use ergodic theory to

produce the pieces out of which an injective surface can be built. I would

like to sketch a proof of the fact that the extension of a free group

associated to a ''random'' endomorphism contains a surface subgroup with

probability one. This is joint work with Alden Walker.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006