Harald Helfgott

Soficity, short cycles and the Higman group

We suggest the beginning of a possible strategy towards finding a non-sofic group, and discuss some candidates. For each "candidate" group G, we show that, if G were sofic, then there would have to be a map from Z/nZ to itself that behaves like an arithmetical function in a local sense and also satisfies a rather strong recurrence property. We also improve on existing bounds on the recurrence of exponential maps on Z/pZ. This is so both in the case where G is the Higman group (joint work with K. Juschenko) and in much greater generality.
Our approach to (non)-soficity is based on the study of sofic representations of amenable subgroups of a sofic group.