Stability results for local zeta functions of groups and related structures

TL;DR

This paper investigates how local zeta functions of groups behave under prime variation and base extensions, revealing that their properties are largely determined by their behavior at almost all primes, with implications for topological zeta functions and functional equations.

Contribution

It demonstrates that the behavior of local zeta functions under prime variation and base extensions can be fully determined by their behavior at a density 1 set of primes, unifying their analysis.

Findings

Behavior under prime variation determines functions for almost all primes.

Explicit formulas relate effects of prime change and base extensions.

Applications include insights into topological zeta functions and functional equations.

Abstract

Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of these operations can be expressed simultaneously in terms of a single explicit formula. We show that in these cases, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.