Combinatorial Commutative Algebra

A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag-Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag-Solitar group by other groups G. We give an elementary condition on G, enjoyed for example by Z ≀ Z and the integral Heisenberg group, under which the resulting group is sofic. We then use soficity to deduce that there exist permutations of Z/nZ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.

To every group G we associate a linear monoidal category Par (G) that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of Par (G) into the group Heisenberg category associated to G. This embedding intertwines the natural actions of both categories on modules for wreath products of G. Finally, we prove that the additive Karoubi envelope of Par (G) is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.