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

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.