Morse Theory and Bestvina–Brady Groups

Section 7.1 develops “Morse theory” for real-valued functions on convex cell complexes. There are two purposes: (1) to study finiteness properties of groups that are defined as the kernel of a homomorphism from a group to the infinite cyclic group, and (2) to study the cohomology of a group with group ring coefficients. In Sect. 7.3 the Bestvina–Brady group BB(L) is defined as the kernel of a surjection from A(L) to the infinite cyclic group \(A_L\to \mathbb {Z}\) , where \(A_L\) is the RAAG associated to a flag complex L. Bestvina and Brady proved that if L is acyclic but not simply connected, then BB(L) is a group that is type FP but not type F. There are various applications. In Sect. 7.4.1 examples are given of \(\operatorname {PD}^n\) groups that are not the fundamental group of any closed aspherical manifold. In Sect. 7.4.2 we describe the examples of Leary–Nucinkis of virtually torsion-free groups G whose torsion-free subgroup is type F, yet G is not type \(\mathcal V\) F. In Sect. 7.4.3 variations of Bestvina–Brady groups are used to get Leary’s construction of uncountably many groups of type FP.