Shift Dynamics for Cyclically Presented Groups

A finite presentation for a group G is 'symmetric' if there is a regular permutation group for the generating set that induces a regular permutation of the relator set, in which case the permutation group acts by automorphisms on G.

I will present a criterion under which the permutation group embeds in the outer automorphism group of G. The criterion involves 'asphericity' of a certain 'relative presentation' associated to the permutation action and which strongly affects the dynamics of the permutation action on the non-identity elements of G. Failure of the criterion implies the existence of combinatorial geometric objects called 'spherical pictures'. I will display some of these diagrams and discuss their algebraic and topological significance.