Constructing thin subgroups of SL(n+1,R) via bending

TL;DR

This paper introduces a method using convex projective geometry to construct numerous new thin subgroups within lattices of SL(n+1,R), expanding the understanding of hyperbolic manifold groups.

Contribution

It demonstrates the construction of infinitely many non-commensurable lattices in SL(n+1,R) containing thin subgroups derived from arithmetic lattices in SO(n,1).

Findings

Constructs new thin subgroups in SL(n+1,R)

Shows existence for a large class of arithmetic lattices

Includes all non-cocompact and even-dimensional cocompact cases

Abstract

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it is possible to find infinitely many non-commensurable lattices in SL(n+1,R) that contain a thin subgroup isomorphic to a finite index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all non-cocompact arithmetic lattices and all cocompact arithmetic lattices when $n$ is even.