Lower bounds for covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$

TL;DR

This paper determines the minimal covolumes of arithmetic lattices in $PSL_2( eal)^n$, identifying unique minimal lattices for various dimensions, extending classical results on Fuchsian groups to higher dimensions.

Contribution

It identifies the unique minimal covolume arithmetic lattices in $PSL_2( eal)^n$ for all $n extgreater=2$, including explicit descriptions and comparisons.

Findings

The Hilbert modular group for the cubic field of discriminant 49 has minimal covolume in $PSL_2( eal)^3$.

The normalizer of a maximal order in a quaternion algebra over the quartic field with discriminant 725 has minimal covolume in $PSL_2( eal)^2$.

These minimal lattices are unique up to conjugation and extend classical minimal covolume results to higher dimensions.

Abstract

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $\mu$ be the Euler-Poincar\'e measure on $PSL_2(\mathbb R)^n$ and $\chi=\mu/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49}})\subset PSL_2(\mathbb R)^3$, with $k_{49}$ the totally real cubic field of discriminant $49$ has the minimal covolume with respect to $\chi$ among all irreducible lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and is unique such lattice up to conjugation. The uniform lattice of minimal covolume with respect to $\chi$ is the normalizer $\Delta_{k_{725}}^u$ of the norm-1 group of a maximal order in the quaternion algebra over the unique totally real quartic field with discriminant $725$ ramified exactly at two infinite places, which is a lattice in $PSL_2(\mathbb R)^2$. There is exactly one more lattice in $PSL_2(\mathbb R)^2$ and exactly one in $PSL_2(\mathbb R)^4$ with the same covolume as $\Delta_{k_{725}}^u$, which are the Hilbert modular groups corresponding to $\mathbb Q(\sqrt{5})$ and $k_{725}$. The two lattices $\Delta_{k_{725}}^u$ and $PSL_2(\mathfrak o_{\mathbb Q(\sqrt{5})})$ have the smallest covolume with respect to the Euler-Poincar\'e measure among all arithmetic lattices in $G_n$ for all $n\geq 2$. These results are in analogy with Siegel's theorem on the unique minimal covolume (uniform and non-uniform) Fuchsian groups and its generalizations to various higher dimensional hyperbolic spaces due to Belolipetsky, Belolipetsky-Emery, Stover and Emery-Stover.