On Higher Graph Manifolds

In this short note we introduce a general notion of 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 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.