On conjugacy growth of linear groups

Let G be a finitely generated linear group. We show that if G has exponential growth (or equivalently is not virtually nilpotent), then its ball of radius n contains exponentially many non-conjugate elements. If in addition G is not virtually solvable, we show that the ball of radius n achieves exponentially many characteristic polynomials. This is joint work with Breuillard, Lubotzky, and Meiri.