Periodic flats and group actions on locally symmetric spaces

TL;DR

This paper demonstrates that finite volume irreducible locally symmetric manifolds of dimension three or higher have maximal symmetry only under their standard locally symmetric metric, and non-locally symmetric metrics have discrete isometry groups on the universal cover.

Contribution

It introduces the use of maximal periodic flats to establish symmetry bounds and characterizes isometry groups of non-locally symmetric metrics on these manifolds.

Findings

No metric on such manifolds exceeds the symmetry of the locally symmetric metric.

Non-locally symmetric metrics lift to universal covers with discrete isometry groups.

Maximal periodic flats are key to understanding symmetry constraints.

Abstract

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discrete isometry group.