On Higher Graph Manifolds

TL;DR

This paper introduces higher graph manifolds, characterizes their volume collapse behavior, computes minimal volume in hyperbolic cases, verifies the coarse Baum--Connes conjecture, and explores scalar curvature and Yamabe invariants.

Contribution

It defines higher graph manifolds, applies barycenter techniques for volume collapse, and provides explicit minimal volume calculations for hyperbolic cases.

Findings

Characterization of volume collapse in higher graph manifolds

Exact minimal volume for hyperbolic pure pieces

Verification of the coarse Baum--Connes conjecture

Abstract

In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and show that they do not admit positive scalar curvature metrics. In the case without any pure pieces, we show the Yamabe invariant vanishes.