Thin surface subgroups of non-uniform arithmetic lattices in $\rm{SO}^+(n,1)$

TL;DR

This paper proves that fundamental groups of certain non-compact, arithmetic hyperbolic manifolds in dimensions four and higher contain thin surface subgroups, and their doubles embed as GFERF subgroups.

Contribution

It establishes the existence of thin surface subgroups in all non-compact, arithmetic hyperbolic n-manifolds for n≥4 and shows doubles embed as GFERF subgroups.

Findings

Fundamental groups of all non-compact, arithmetic hyperbolic n-manifolds contain thin surface subgroups.

Doubles of cusped, arithmetic hyperbolic n-manifolds embed as GFERF subgroups.

The results hold for dimensions n≥4.

Abstract

We show that the fundamental groups of all non-compact, arithmetic, hyperbolic, $n$-manifolds for $n\geq 4$ contain thin surface subgroups. As a consequence of the proof of this theorem we also show that the fundamental groups of the doubles of cusped, arithmetic, hyperbolic $n$-manifolds embed as GFERF subgroups of $\rm{SO}^+(n+1,1)$.