Counting flat cycles in the homology of locally symmetric spaces

TL;DR

This paper establishes lower bounds on the contribution of immersed flat manifolds to the homology of congruence covers of locally symmetric spaces, enhancing understanding of their topological structure.

Contribution

It provides new lower bounds for the homological contribution of flat cycles in congruence covers of locally symmetric spaces, extending previous results to broader families.

Findings

Lower bounds for flat cycle contributions in homology

Results apply to various locally symmetric spaces

Advances understanding of topological complexity

Abstract

Locally symmetric spaces like $SL(n,\mathbb Z)\backslash SL_n(\mathbb R)/SO(n)$ contain immersed compact flat manifolds of dimension equal to the real rank. We give a lower bound for the contribution of these cycles to the homology of congruence covers. Similar results are proved for other families of locally symmetric spaces.