Systolic lattice extensions of classical Schottky groups

TL;DR

This paper constructs systolic lattice extensions of classical Schottky groups in hyperbolic spaces, demonstrating density of complex translation lengths in hyperbolic manifolds and offering new methods for non-arithmetic lattice construction.

Contribution

It introduces systolic lattice extensions of classical Schottky groups and explores their implications for hyperbolic manifold translation lengths and lattice construction.

Findings

Density of systoles' complex translation lengths in hyperbolic d-manifolds for all d≥3

Construction of non-arithmetic lattices via new methods

Extension of classical Schottky groups to lattice extensions

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.