Geometry of locally compact groups of polynomial growth and shape of large balls

Abstract

We generalize Pansu’s thesis [27] about asymptotic cones of finitely generated nilpotent groups to arbitrary locally compact groups $G$ of polynomial growth. We show that any such $G$ is weakly commensurable to some simply connected solvable Lie group $S$, the Lie shadow of $G$ and that balls in any reasonable left invariant metric on $G$ admit a well-defined asymptotic shape. By-products include a formula for the asymptotics of the volume of large balls and an application to ergodic theory, namely that the Ergodic Theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and give a geometric proof of some results of Stoll [33] of the rationality of growth series of Heisenberg groups.