Randomness, genericity, and ubiquity of hyperbolic behavior in groups.

Consider an infinite group G acting by isometries on some metric space X.
How does a “typical” element act? Consider a representation of G into some matrix group. What sort of matrix represents “typical” elements of G?

The answer depends on what we mean by the word “typical,” of which there are at least two reasonable notions. We may take a random walk on G and look where it lands after a large number of steps. We may also fix a generating set for G and look how large balls are distributed.

I will talk about how these two notions of genericity are related and how they differ, focusing on the setting of hyperbolic groups. I will also explain that the following is true with respect to both notions: For a group acting on a Gromov hyperbolic metric space typical elements act loxodromically, i.e. with north-south dynamics.

For a representation of a large class of groups (including hyperbolic groups) into SL_n R, typical elements map to matrices whose eigenvalues are all simple and have distinct moduli.