Conjugacy Growth Series of Some Infinitely Generated Groups

It is observed that the conjugacy growth series of the infinite finitary 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. Pour le parfait flâneur, pour l’observateur passionné, c’est une immense jouissance que d’élire domicile dans le nombre, dans l’ondoyant dans le mouvement, dans le fugitif et l’infini. (Baudelaire, in Le peintre de la vie moderne [Baud–63].) 1. Explicit conjugation growth series Let G be a group generated by a set S. For g ∈ G, the word length lG,S(g) is defined to be the smallest non-negative integer n for which there are s1, s2, . . . , sn ∈ S ∪ S such that g = s1s2 · · · sn, and the conjugacy length κG,S(g) is the smallest integer n for which there exists h in the conjugacy class of g such that lG,S(h) = n. For n ∈ N, denote by γG,S(n) ∈ N∪{∞} the number of conjugacy classes in G consisting of elements g with κG,S(g) = n (we agree that Date: June 15, 2016. 2000 Mathematics Subject Classification. 20F69, 20F65.