Thin subgroups isomorphic to Gromov--Piatetski-Shapiro lattices

TL;DR

This paper constructs numerous examples of thin subgroups within special linear groups, isomorphic to fundamental groups of non-arithmetic hyperbolic manifolds, by embedding Gromov--Piatetski-Shapiro lattices into SL(n+1,R).

Contribution

It demonstrates how to embed non-arithmetic lattices from hyperbolic geometry into linear groups as thin subgroups, expanding understanding of their structure.

Findings

Gromov--Piatetski-Shapiro lattices can be embedded into SL(n+1,R).

Embedded images are thin subgroups.

Provides new examples of thin subgroups in linear groups.

Abstract

In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $\mathrm{SO}(n,1)$ constructed by Gromov and Piatetski-Shapiro can be embedded into $\mathrm{SL}_{n+1}(\mathbb{R})$ so that their images are thin subgroups