Discreteness of deformations of cocompact discrete subgroups

We prove the discreteness of small deformations of a discrete cocompact subgroup of isometries of a locally compact metric space under some natural restrictions.

Crystallographic groups with trivial center and outer automorphism group

Let Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

On Elliptic Curves in SL2(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

Let Γ be a discrete cocompact subgroup of SL2(ℂ). We conjecture that the quotient manifold X = SL2(ℂ) / Γ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel’s conjecture holds. We also prove it in the special case where Γ ∩ SL2(∝) is cocompact in SL2(ℝ).Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.