The fundamental group of a toroidal compactification of a Hermitian locally symmetric space

TL;DR

This paper determines the fundamental group of a toroidal compactification of a Hermitian symmetric space quotient and explores conditions for residual finiteness of this group, linking topology with lattice properties.

Contribution

It explicitly computes the fundamental group of the compactification and establishes conditions for residual finiteness in the context of non-uniform lattices.

Findings

Fundamental group of X' equals that of X.

Ranks of first homology groups are equal.

Residual finiteness condition for the fundamental group.

Abstract

The present work obtains the fundamental group of a toroidal compactification X' of a non-compact quotient X of a Hermitian symmetric space D of non-compact type by a lattice L in the isometry group G of D. As a consequence it derives the equality of the ranks of the first homology groups of X' and X with integral coefficients. The paper provides also a sufficient condition on a torsion free non-uniform lattice L, under which the fundamental group of X' is residually finite. Articles of Hummel-Schroeder, Hummel and Di Cerbo imply that the toroidal compactifications X' of generic non-compact torsion free quotients X of the complex balls satisfy this sufficient condition.