Families of Polynomial Mappings Between Groups

In 2004-2005, A. Leibman introduced the concept of polynomial sequences in a group and, more generally, of polynomial mappings between groups. These polynomial mappings are generalizations of group homomorphisms roughly in the same sense that higher-degree ordinary polynomials generalize linear functions.

In joint ongoing work with J. Iovino, we introduce the new concept of a family of polynomial mappings between groups. These families (which are themselves groups) encompass and generalize Leibman's classes of polynomial mappings, while removing Leibman's a priori assumption that the codomain is a nilpotent group and also having significant potential for ergodic applications. (For instance, in 2012 M. Walsh generalized von Neuman's classical ergodic theorem to the case of nilpotent polynomial group sequences.) The purely algebraic aspects of the study of families of polynomial mappings are elementary, and we pose several open questions suitable as undergraduate research topics.