Generalized Graph Manifolds, Residual Finiteness, and the Singer Conjecture

TL;DR

This paper proves the Singer conjecture for certain classes of graph manifolds with residually finite fundamental groups, providing new insights into their geometric and algebraic properties.

Contribution

It establishes the Singer conjecture for extended and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups, and identifies classes of higher graph manifolds with this property.

Findings

Proves Singer conjecture for extended graph manifolds.

Proves Singer conjecture for pure complex-hyperbolic higher graph manifolds.

Identifies classes of higher graph manifolds with residually finite fundamental groups.

Abstract

We prove the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups. In real dimension three, where a result of Hempel ensures that the fundamental group is always residually finite, we then provide a Price type inequality proof of a well-known result of Lott and Lueck. Finally, we give several classes of higher graph manifolds which do indeed have residually finite fundamental groups.