Combination Theorems and the Farrell--Jones Conjecture for Graph Products of Pairs

Abstract

We study the Farrell--Jones conjecture for graph products of pairs and for fundamental groups of polyhedral products. We introduce a pair-deletion decomposition and a local notion of FJ-admissibility, and prove a vertexwise combination theorem for graph products of pairs. The criterion applies, in particular, to ordinary graph products, free-factor pairs, suitable normal pairs, and classes controlled by acylindrical Bass--Serre actions. For polyhedral products, we combine these results with their description by relative graph products and graph products of pairs. We also obtain bounds and exact computations for amenable category, together with applications to simplicial volume and Borel rigidity of closed aspherical polyhedral-product manifolds.