Systolic lattice extensions of classical Schottky groups

Abstract

We produce lattice extensions of a dense family of classical Schottky subgroups of the isometry group of $d$-dimensional hyperbolic space. The extensions produced are said to be systolic, since all loxodromic elements with short translation length are conjugate into the Schottky groups. Various corollaries are obtained, in particular showing that for all $d\geq3$, the set of complex translation lengths realized by systoles of closed hyperbolic $d$-manifolds is dense inside the set of all possible complex translation lengths. We also consider complex translation lengths in arithmetic hyperbolic $d$-manifolds, and provide a new way to construct non-arithmetic lattices.