# Counting non-uniform lattices

## TL;DR

This paper investigates the asymptotic growth of non-uniform lattices in certain Lie groups, confirming a conjecture for most cases and refining understanding of lattice counting in higher rank groups.

## Contribution

It proves that the conjectured growth rate holds for non-uniform lattices in most simple Lie groups of real rank at least 2.

## Key findings

- Confirmed the conjectured asymptotic growth for non-uniform lattices in most groups.
- Disproved the original conjecture for all lattices, but verified it for non-uniform cases.
- Provided explicit constants related to the root systems of the groups.

## Abstract

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices.