A group is sofic when every finite subset can be well approximated in a finite symmetric group. The outstanding question, due to Gromov, is whether every group is sofic.
Helfgott and Juschenko argued that a celebrated group constructed by Higman is unlikely to be sofic because its soficity would imply the existence of some seemingly pathological functions.  I will describe joint work with Martin Kassabov and Vivian Kuperberg in which we construct variations on Higman's group and explore their soficity.