Addendum to: On volumes of arithmetic quotients of SO (1, n)

Abstract

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments. Mathematics Subject Classification (2000): 11E57 (primary); 22E40 (secondary). 1.1. Let us recall some notation and basic notions. Following [1] we will assume that n is even and n ≥ 4. The group of orientation preserving isometries of hyperbolic n-space is isomorphic to SO(1, n)o, the connected component of the identity of the special orthogonal group of signature (1, n), which can be identified with SO0(1, n), the subgroup of SO(1, n) preserving the upper half space. This group is not Zariski closed in SLn+1 thus in order to construct arithmetically defined subgroups of SO(1, n)o we consider arithmetic subgroups of the orthogonal group SO(1, n) or, more precisely, of groups G = SO( f ) where f is an admissible quadratic form defined over a totally real number field k (see [1, Section 2.1]). We have an exact sequence of k-isogenies: 1 → C → G φ → G → 1, (1.1) where G(k) Spin( f ) is the simply connected cover of G and C μ2 is the center of G. This induces an exact sequence in Galois cohomology (see [5, Section 2.2.3]) G(k) φ → G(k) δ → H1(k, C) → H1(k, G). (1.2) The main idea of this note is that by using (1.2) certain questions about arithmetic subgroups of G can be reduced to questions about the Galois cohomology group H1(k, C). A coherent collection of parahoric subgroups P = (Pv)v∈V f of G (V f = V f (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Received October 16, 2006; accepted in revised form March 12, 2007.