Counting arithmetic lattices and surfaces

TL;DR

This paper estimates the number of arithmetic lattices and surfaces in simple Lie groups, providing concrete asymptotic results especially for 3-manifolds and classical groups like PSL(2,R).

Contribution

It offers the first explicit asymptotic estimate for the count of arithmetic 3-manifolds and computes the growth rate for lattices in PSL(2,R).

Findings

Asymptotic growth rate of arithmetic lattices in PSL(2,R) determined.

First concrete estimate on the number of arithmetic 3-manifolds.

Multiple techniques used: geometric, number theoretic, and character value estimates.

Abstract

We give estimates on the number $AL_H(x)$ of arithmetic lattices $\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\Gamma$).