2/15/2005

Emmanuel Breuillard,
IHES

The asymptotic shape of metric balls in groups of polynomial growth, and pointwise ergodic theorems

Let $G$ be a locally compact group of polynomial growth, i.e.vol(U^n)=O(n^K) for some K>0 and some compact generating set U. We show that there is a number c(U)>0 and an integer d that can be computed explicitely such that the n^(-d)vol(U^n) converges to c(U) as n tends to infinity. We give a geometric interpretation of the asymptotic volume c(U) as the volume of the unit ball for some Carnot-Caratheodory metric on a stratified simply connected nilpotent Lie group naturally associated to G. As a consequence we get in particular that the balls {U^n}_n form a Folner sequence, thus answering Greenleaf's "localization" problem in this case. The results hold for a large class of other metrics on G, such as left invariant Riemannian metrics if G is Lie, and enable to prove pointwise ergodic theorems for such averages, thus answering a question of A. Nevo.