# Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls

## TL;DR

This paper studies the asymptotic volume and shape of large balls in locally compact groups with polynomial growth, revealing geometric structures and convergence properties, with applications to ergodic theory and group theory.

## Contribution

It generalizes Pansu's thesis to arbitrary locally compact groups, showing their geometric relation to solvable Lie groups and describing the asymptotic shape of large balls.

## Key findings

- Large balls in such groups have a well-defined asymptotic shape.
- Any such group is weakly commensurable to a simply connected solvable Lie group.
- Results include applications to ergodic theory and answers to open questions.

## Abstract

We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.