Variations on the theme of zeta functions of groups and rings - Lee, S (2019)
In recent years the study of zeta functions of groups and rings has been expanded rapidly. Many new zeta functions enumerating different subobjects have been introduced and studied. However, we still encounter difficulties when one wants to explicitly compute them. This thesis comprises three parts. The first part contains the computation of ideal zeta functions for low dimensional, class-2 nilpotent Lie rings. In addition, we discuss connections to Higman's PORC conjecture. The second part, which is the main part of the thesis, consists of the various approximations of zeta functions of groups and rings, namely: graded ideal zeta functions (Chapter 3 and 4), finite zeta functions enumerating subalgebras (or ideals) of finite Lie Fp-algebras over Fp (Chapter 5), and the i-th partial zeta functions of groups and rings (Chapter 6). We provide explicit calculations in many cases, where the classic subgroup/normal subgroup zeta functions are not available. In addition we study properties of the zeta functions obtained, such as uniformity properties or the existence of functional equations. In the final part of the thesis we are concerned with the connection between the study of zeta functions of groups and the study of finite p-groups. In particular we show how our methods from zeta functions of groups and rings can be adapted to study finite p-groups. With these new approaches we end our thesis with several interesting conjectures. Most of the explicit calculations in this thesis have been checked and verified with computer algebra system MAGMA.