Conjugacy growth series of some infinitely generated groups

TL;DR

This paper explores the conjugacy growth series of various infinite and finite groups, revealing connections to partition functions, congruences, and introducing the concept of partition-complete graphs.

Contribution

It computes conjugacy growth series for several groups and generating sets, introduces partition-complete graphs, and proposes generalized Ramanujan congruences with supporting numerical evidence.

Findings

Conjugacy growth series of the infinite symmetric group relate to the partition function.

Identification of partition-complete graphs generalizing semi-hamiltonian graphs.

Evidence for generalized Ramanujan congruences in group-related series.

Abstract

It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural "generalized Ramanujan congruences", as we call them, for which we give numerical evidence in Appendix C.