Incompressible surfaces in rank one locally symmetric spaces

TL;DR

This paper proves that most cocompact lattices in rank one simple Lie groups, except for a specific case, contain surface subgroups, advancing understanding of their geometric subgroup structure.

Contribution

It establishes the existence of surface subgroups in cocompact lattices of rank one simple Lie groups, excluding SO(2m,1).

Findings

Cocompact lattices in most rank one simple Lie groups contain surface subgroups.

The exception is the group SO(2m,1).

Provides new insights into subgroup structures of these lattices.

Abstract

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.