Conjugacy growth series for wreath product finitary symmetric groups

TL;DR

This paper introduces new conjugacy growth series for finitary alternating wreath products linked to mixed weight modular forms, demonstrating the existence of prime-power congruences and proposing a general method for their study.

Contribution

It defines a novel family of conjugacy growth series related to mixed weight modular forms and establishes their congruence properties, extending previous work on permutation groups.

Findings

Congruences exist modulo powers of all primes p ≥ 5.

A method for studying congruences of sums of mixed weight modular forms is proposed.

New conjugacy growth series are connected to modular forms of mixed weights.

Abstract

In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of these series. We define a new family of conjugacy growth series for the finitary alternating wreath product that are related to sums of modular forms of integer and half-integral weights, the so-called \textit{mixed weight modular forms}. The previous works motivate the study of congruences for these series. We prove that congruences exist modulo powers of all primes $p \geq 5$. Furthermore, we lay out a method for studying congruence properties for sums of mixed weight modular forms in general.