If H is an almost normal subgroup of G, both are of type VFP, and vcd(G)=vcd(H)+1, then G is the fundamental group of a finite graph of groups with vertex and edge groups commensurable to H.
Bridson and Andr \'e Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol
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Groups of cohomological codimension one
If H is an almost normal subgroup of G, both are of type VFP, and vcd(G)=vcd(H)+1, then G is the fundamental group of a finite graph of groups with vertex and edge groups commensurable to H.