Fundamental groups of toroidal compactifications

TL;DR

This paper calculates the fundamental group of toroidal compactifications of Hermitian locally symmetric spaces without restrictions on the lattice, and provides bounds for their first Betti number.

Contribution

It offers the first computation of the fundamental group for these compactifications without assuming neatness or arithmeticity.

Findings

Fundamental group computed for general lattices.

Bounds established for the first Betti number.

Results apply to a broad class of Hermitian locally symmetric spaces.

Abstract

We compute the fundamental group of a toroidal compactification of a Hermitian locally symmetric space $D/\Gamma$, without assuming either that $\Gamma$is neat or that it is arithmetic. We also give bounds for the first Betti number.