Residual properties of graph manifold groups

TL;DR

This paper investigates the residual properties of graph manifold groups, correcting a previous assumption and extending the main results to a broader class of groups with implications for 3-manifold topology.

Contribution

It corrects a misconception about residual properties of graph manifold groups and adapts existing proofs to establish new residual properties for these groups.

Findings

Not all graph manifold groups are residually p for every prime p.

Modified arguments recover the main topological result of Perron and Shalen.

Characterized all semidirect products Z ⋉ Z^d that are residually p for all primes.

Abstract

Let $f\colon M\to N$ be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen showed that if $f$ induces a homology equivalence on all finite covers, then $f$ is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a $3$-manifold whose fundamental group is residually $p$ for every prime $p$. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products $\Z \ltimes \Z^d$ which are residually $p$ for every prime $p$.