Emma Smith

Distinct Difference Configurations in Groups

A subset D of a group is a Distinct Difference Configuration (DDC) if all differences g^{-1} h, are distinct elements of D, are distinct. For a finitely-generated infinite group, the DDC growth problem asks: how large can a DDC be if it is contained in a ball of radius r?

After introducing DDCs and this growth problem, I will present a probabilistic lower bound showing that large DDCs exist in groups where involutions grow slowly. Applied to the Fabrykowski-Gupta group, this demonstrates an example where the maximal size of a DDC exhibits intermediate (superpolynomial but subexponential) growth. I will also preset sharper results for free groups, where explicit constructions can be extended to groups containing free subgroups such as SL(2,Z).