Height Gap, an Arithmetic Margulis Lemma and Almost Laws

Abstract: We provide a new (more elementary) proof of a result of E. Breuillard, which state that a set of matrices with algebraic entries generating a non-virtually solvable group has a positive lower bound in its arithmetic height (we will explain this notion), this is a non-abelian version of Lehmer’s problem. We also show that in arithmetic locally symmetric spaces, short geodesics tend to be far from each other if the degree of the trace field is large. This lemma allows us to prove new results about growth of cohomology of sequences of locally symmetric spaces and to give a proof of a conjecture of Gelander. These results are works in progress with Joe Chen and Homin Lee, and with Mikolaj Fraczyk and Jean Raimbault.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar