Zeta functions of groups and rings – functional equations and analytic uniformity

Abstract

Zeta functions are widely used tools in the study of asymptotic properties of infinite groups and rings, in particular their subobject and representation growth. We survey recent results on arithmetic and asymptotic features of such functions, focussing on various classes of subobject zeta functions, in particular submodule zeta functions associated with nilpotent algebras of endomorphisms, and representation zeta functions associated to arithmetic groups, specifically finitely generated nilpotent groups.