# Asymptotic invariants of lattices in locally compact groups

## TL;DR

This paper investigates the asymptotic behavior of sequences of lattices in a fixed locally compact group, focusing on Betti numbers, rank gradient, and their relation to ultraproducts of group actions.

## Contribution

It introduces a novel approach using ultraproducts of group actions to analyze the asymptotic invariants of lattices in locally compact groups.

## Key findings

- Asymptotic growth of Betti numbers normalized by covolume.
- Relationship between properties of ultraproduct cross sections and lattice invariants.
- Insights into the rank gradient behavior in lattice sequences.

## Abstract

The aim of this work is to understand some of the asymptotic properties of sequences of lattices in a fixed locally compact group. In particular we will study the asymptotic growth of the Betti numbers of the lattices renormalized by the covolume and the rank gradient, the minimal number of generators also renormalized by the covolume. For doing so we will consider the ultraproduct of the sequence of actions of the locally compact group on the coset spaces and we will show how the properties of one of its cross sections are related to the asymptotic properties of the lattices.