Infinitely many hyperbolic Coxeter groups through dimension 19

Abstract

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact groups for n < 7. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.Comment: This is the version published by Geometry & Topology on 11 July 2006 (V2: typesetting correction

Local rigidity in quaternionic hyperbolic space

“…There is an easy manner to deform the embedding ρ 0 : Γ → Sp(1, 1) → Sp(2, 1). Indeed, since Sp(2, 1) contains Sp(1, 1) × Sp(1), it also contains many copies of Sp(1, 1) × U (1). If H 1 (Γ, R) = 0, which happens sometimes (see [17]), the trivial representation Γ → U(1) can be continuously deformed to a nontrivial representation ρ 1 .…”

“…It is highly expected that such a global rigidity should hold in quaternionic hyperbolic spaces, but we have been unable to prove it. Note that since Sp(1, 1) = Spin (4,1) 0 , there exist uniform lattices in Sp(1, 1) which are isomorphic to Zariski dense subgroups of Sp(4, 1), see section 7.…”

“…Let Γ ⊂ Sp(1, 1) be a uniform lattice. Embed Γ into Sp (3,1). Can one deform Γ to a Zariski dense subgroup?…”

“…Theorem 1.2. Let Γ ⊂ Spin (3,1) 0 be a lattice. Embed Spin(3, 1) 0 into Spin(4, 1) 0 = Sp(1, 1) and then into Sp(2, 1) in the obvious manner.…”

“…If the assumption on parabolics is removed, nonuniform lattices in Spin (3,1) 0 can be deformed within Spin(3, 1) 0 , see [21], chapter 5. Question.…”