Comparison of Volumes of Siegel Sets and Fundamental Domains of $\mathrm{SL}_n(\mathbb{Z})$

TL;DR

This paper explicitly computes and compares the volumes of Siegel sets and fundamental domains for $ ext{SL}_n( ext{Z})$ in $ ext{SL}_n( ext{R})$, revealing super-exponential growth in their volume ratio as $n$ increases.

Contribution

It provides explicit volume calculations for Siegel sets and fundamental domains, and analyzes their ratio, offering new bounds and insights into their asymptotic behavior.

Findings

Volume ratio grows super-exponentially with $n$

Super-exponential lower bound for intersecting Siegel sets

Upper bounds for the number of intersections

Abstract

The purpose of this paper is to calculate explicitly the volumes of Siegel sets which are coarse fundamental domains for the action of $\mathrm{SL}_n (\mathbb{Z})$ in $\mathrm{SL}_n (\mathbb{R})$, so that we can compare these volumes with those of the fundamental domains of $\mathrm{SL}_n (\mathbb{Z})$ in $\mathrm{SL}_n (\mathbb{R})$, which are also computed here, for any $n\geq 2$. An important feature of this computation is that it requires keeping track of normalization constants of the Haar measures. We conclude that the ratio between volumes of fundamental domains and volumes of Siegel sets grows super-exponentially fast as $n$ goes to infinity. As a corollary, we obtained that this ratio gives a super-exponencial lower bound, depending only on $ n $, for the number of intersecting Siegel sets. We were also able to give an upper bound for this number, by applying some results on the heights of intersecting elements in $ \mathrm{SL}_n (\mathbb{Z}) $.