Dynamical Induction and Cohomology of Crystallographic Groups

Adem, Lueck, and their collaborators have shown that in various situations, the Lyndon-Hochschild-Serre spectral sequence associate to a crystallographic group collapses at the second page. Crystallographic groups are built from integral representations finite groups. We describe a generalized notion of induction, which we call dynamical induction, originating in the theory of C*-algebras, and we show that these collapse results are preserved under induction in an appropriate sense. In particular, we show that the spectral sequence associated to a semidirect product of the form Z^n \rtimes Q collapses whenever the action of Q on Z^n is induced up from an action of a group of square-free order. This is joint work with my recent Ph.D. student, Chris Neuffer.