ADDENDUM TO: ON VOLUMES OF ARITHMETIC QUOTIENTS OF SO(1, n)

There are errors in the proof of the 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 give several remarks on further development. 1.1. Let us recall some notations and basic notions. Following [B] we will assume that n is even. The group of orientation preserving isometries of the hyperbolic nspace is isomorphic to SO(1, n), the connected component of 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) 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 [B, Sect. 2.1]). We have an exact sequence of k-isogenies: (1) 1 → C → G̃ φ → G → 1, where G̃(k) ≃ Spin(f) is the simply connected cover of G and C ≃ μ2 is the center of G̃. This induces an exact sequence in Galois cohomology (2) G̃(k) φ → G(k) δ → H(k,C) → H(k, G̃). The main idea of this note is that using (2) certain questions about arithmetic subgroups of G can be reduced to the Galois cohomology groups H(k,C). A coherent collection of parahoric subgroups P = (Pv)v∈Vf of G̃ (Vf = Vf (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Λ = G̃(k) ∩ ∏ v∈Vf Pv ⊂ G̃(k) (see [BP]). We fix an infinite place v of k for which G(kv) ≃ SO(1, n) and denote it by Id. The image of Λ under the central kisogeny φ is an arithmetic subgroup of G and every maximal arithmetic subgroup of G(kId) can be obtained as a normalizer of some φ(Λ) [BP, Prop. 1.4]. We will also consider the local stabilizers of P in the adjoint group G(= G), defining P v to be the stabilizer of Pv in G(kv) and P = (P v)v∈Vf . Clearly, P v ⊃ φ(Pv). In the notation of [B] the subgroups φ(Pv) are called parahoric subgroups of G, however this terminology is non-standard and we will avoid using it here. 1991 Mathematics Subject Classification. 11E57 (primary); 22E40 (secondary).