Conference: Spring Lectures 2012


Homogeneous flows, Invariant measures and divisibility

{{_Ltalk:R}} Prof. Dr. Shahar Mozes
Date: 04.05.12   Time: 14.30 - 15.20   Room: Y27H28

The interaction between homogeneous flows and number theoretical questions has a long and fruitful history. A celebrated example is the relation between the dynamics of actions of subgroups generated by unipotents and the Oppenheim conjecture concerning values of quadratic forms. Another family of homogeneous flows which has been studied extensively in recent years is actions by diagonalizable subgroups. In a joint work with Manfred Einsiedler we study a relationship between the dynamical properties of the action a maximal diagonalizable group $A$ on certain arithmetic quotients $G/\Gamma$ where $G$ is a Lie group and $\Gamma<G$ a lattice, and arithmetic properties of the lattice. In particular, given a finite set of odd primes with at least two elements we consider the semigroup of all integer quaternions that have norm equal to a product of powers of primes from the set. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given quaternion from the semigroup has subexponential growth.