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

Abstract

We study the covolumes of arithmetic lattices in PSL2(R) n for n ≥ 2 and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let µ be the Euler-Poincaré measure on PSL2(R) n and χ = µ/2 n . We show that the Hilbert modular group PSL2(o k 49 ) ⊂ PSL2(R) 3 , with k49 the totally real cubic field of discriminant 49 has the minimal covolume with respect to χ among all irreducible lattices in PSL2(R) n for n ≥ 2 and is unique such lattice up to conjugation. The cocompact lattice of minimal covolume with respect to χ is the normalizer ∆ u k 725 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 PSL2(R) 2 . There is exactly one more lattice in PSL2(R) 2 and exactly one in PSL2(R) 4 with the same covolume as ∆ u k 725 , which are the Hilbert modular groups corresponding to Q( √ 5) and k725. The two lattices ∆ u k 725 and PSL2(o) ) have the smallest covolume with respect to the Euler-Poincaré measure among all arithmetic lattices in Gn for all n ≥ 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.