The asymptotic shape of metric balls in Lie groups of polynomial

Seminar: 
Group Actions and Dynamics
Event time: 
Monday, January 31, 2005 - 11:30am to Sunday, January 30, 2005 - 7:00pm
Location: 
431 DL
Speaker: 
Emmanuel Breuillard
Speaker affiliation: 
IHES
Event description: 

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 enormalization, 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. This answers a question of A. Nevo and generalizes results of P. Pansu for discrete finitely generated nilpotent groups. As an application, we also derive new pointwise ergodic theorems on nilpotent Lie groups and Lie groups of polynomial growth.