Soficity, short cycles, and the Higman group

Abstract

We suggest the beginning of a possible strategy towards finding a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from Z/pZ to itself, locally like an exponential map, satisfying a rather strong recurrence property. We also improve on existing bounds on the recurrence of exponential maps on Z/pZ. Our approach to (non)-soficity is based on the study of sofic representations of amenable subgroups of a sofic group.