On Higher Graph Manifolds

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.

23 pages; v2 Final version, corrected mistakes in some proofs, results remain the same

References in corpus (3)

• Entropies, volumes, and Einstein metrics
• Smooth (non)rigidity of cusp-decomposable manifolds
• Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

Cited by in corpus (3)

• Positivity of Simplicial Volume for Nonpositively Curved Manifolds with a Ricci-type curvature condition
• Some remarks on the simplicial volume of nonpositively curved manifolds
• On the quasi-isometric rigidity of chambers and walls in cusp-decomposable manifolds