Math Seminars.
Emmanuel Breuillard (Lille 1)
The asymptotic shape of metric balls in groups of polynomial growth
Let $G$ be a connected Lie group of polynomial growth. We show
that $G$ has strict polynomial growth and obtain a formula for the
asymptotics of the volume of large balls. This is done via the study of
the asymptotic shape of metric balls. We show that large balls, after a
suitable renormalization, converge to a limiting compact set, which can be
interpreted geometrically as the unit ball for some Carnot-Caratheodory
metric on the associated graded nilshadow. The results hold for a large
class of pseudometrics including left invariant Riemannian metrics or
``word metrics'' associated to a compact generated set. A similar
description can be made in the greater generality of locally compact $G$.
As an application, we derive new pointwise ergodic theorems on nilpotent
Lie groups and Lie groups of polynomial growth.