Zeta Functions of Groups and Rings – Recent Developments

I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations. 1. About these notes Over the last few decades, zeta functions have become important tools in various areas of asymptotic group and ring theory. With the first papers on zeta functions of groups published barely 25 years ago, the subject area is still comparatively young. Recent developments have led to a wealth of results and gave rise to new perspectives on central questions in the field. The aim of these notes is to introduce the nonspecialist reader informally to some of these developments. I concentrate on two types of zeta functions: firstly, zeta functions in subgroup and subring growth of infinite groups and rings, enumerating finite-index subobjects. Secondly, representation zeta functions in representation growth of infinite groups, enumerating finite-dimensional irreducible complex representations. I focus on common features of these zeta functions, such as Euler factorizations, local functional equations, and their behaviour under base extension. Subgroup growth of groups is a relatively mature subject area, and the existing literature reflects this: zeta functions of groups feature in the authoritative 2003 monograph [39] on “Subgroup Growth”, are the subject of the Groups St Andrews 2001 survey [16] and the report [18] to the ICM 2006. The book [21] contains, in particular, a substantial list of explicit examples. Some more recent developments are surveyed in [32, Chapter 3]. On the other hand, few papers on representation zeta functions of infinite groups are older than ten years. Some of the lecture notes in [32] touch on the subject. The recent survey [31] on representation growth of groups complements the current set of notes. In this text I use, more or less as blackboxes, the theory of p-adic integration and the Kirillov orbit method. The former provides a powerful toolbox for the treatment of a number of group-theoretic counting problems. The latter is a general method to parametrize the irreducible complex representations of certain groups in terms of coadjoint orbits. Rather than to explain in detail how these tools are employed I will refer to specific references at appropriate places in the text. I all but ignore the rich subject of zeta functions enumerating representations or conjugacy classes of finite groups of Lie type; see, for instance, [34]. These notes grew out of a survey talk I gave at the conference Groups St Andrews 2013 in St Andrews. I kept the informal flavour of the talk, preferring instructive examples and sample theorems over the greatest generality of the presented results. As 2010 Mathematics Subject Classification. Primary: 22E50, 20E07. Secondary: 22E55, 20F69, 22E40, 20C15, 20G25, 11M41.