Discrete Linear Groups containing Arithmetic Groups

TL;DR

This paper proves that Zariski dense discrete subgroups of simple real algebraic groups containing higher rank lattices are essentially lattices themselves, with specific results for subgroups of SL_n(R) containing SL_3(Z).

Contribution

It establishes new conditions under which discrete subgroups containing higher rank lattices are actually lattices, extending previous understanding of their structure.

Findings

Zariski dense subgroups with higher rank lattices are lattices in G

Subgroups of SL_n(R) containing SL_3(Z) are conjugate to SL_n(Z)

Provides criteria for when discrete subgroups are lattices in simple algebraic groups

Abstract

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group $G$ which contains a higher rank lattice is a lattice in the group $G$. For example, we show that a Zariski dense subgroup of $SL_n({\mathbb R})$ which contains $SL_3({\mathbb Z})$ in the top left hand corner, is conjugate to $SL_n({\mathbb Z})$ .