We will survey a series of recent developments in the area of first-order descriptions of linear groups. The goal is to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups. We also dwell on the principal problem of isotipicity of finitely generated groups.
I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder – the primitivity rank π(w), the smallest rank of a subgroup of F containing w as an imprimitive element. A sample application is that every subgroup of G of rank less than π(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.
This video is part of the New York Group Theory Cooperative‘s Manhatten algebra day.
