Representation Growth
In 1988, Grunewald, Segal, and Smith [3] initiated the study of subgroup growth: let G be a finitely generated group, and denote by an(G) the number of subgroups of G of index n. (This is always a finite number.) The sequence of numbers an(G) (and their partial sums sn(G) ) encode a lot of arithmetic information about the group structure. One of the highlights of this theory is the following result.
Theorem. Let G be a finitely generated, residually finite group. The sequence sn(G) is bounded by a polynomial if and only if G is virtually soluble of finite rank.
Another sequence of numbers that one may define is the number rn(G) of (inequivalent) irreducible complex representations of G, whose kernels have finite index. (If G carries the structure of a topological group, then the kernels are required to be closed subgroups, so that the representation is continuous.) In this case it is not clear that all rn(G) are in fact finite. Indeed, all rn(G) are finite if and only if all subgroups of finite index have finite abelianization.
Whereas the an(G) may be constant (for example, an(G)=1 if G is infinite cyclic), this is not true for the numbers rn(G), as demonstrated by the following theorem.
Theorem (Craven [1], Jaikin-Zapirain [4], Moretó [5]: see [2]). Let G be a finitely generated group, such that the numbers rn(G) are finite for all n. Then the following are equivalent:
(i) the finite residual of G has finite index; and
(ii) the numbers rn(G) are bounded by such integer N for all n.
In particular, if G is residually finite and the rn(G) are universally bounded, then G is finite.
Considering the case where some of the rn(G) are infinite, let I(G) denote the set of all i such that ri(G) is infinite. In the case where the set I(G) is non-empty but finite, it can be shown that G is virtually abelian. Thus, we get the following partition of the set of finitely generated, residually finite groups.
Theorem (Craven [2]). Let G be a finitely generated, residually finite group. One of the following holds.
(i) I(G) is empty, and all but finitely many of the rn(G) are zero. In this case, G is finite.
(ii) I(G) is empty, the sequence rn(G) is unbounded. In this case, G is a FAb group.
(iii) I(G) is non-empty but finite. In this case, G is virtually abelian.
(iv) I(G) is infinite.
In [2], an explicit lower bound for the maximum of rn(Sm) is obtained. It is of the order of m0.16, which is of course a rational function in m. This forms the basis of the following conjecture.
Conjecture (Craven [2]). Let m(n) denote the maximal multiplicity of the character degrees for the symmetric group Sn. Then there are real numbers a and b such that
na< m(n)<nb
for all sufficiently large n.
References
[1] David A. Craven, Symmetric Group Character Degrees and Hook Numbers, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 26–50. MathSciNet, PDF
[2] David A. Craven, Lower Bounds for Representation Growth, J. Group Theory 13 (2010), no. 6, 873–890. DOI, PDF
[3] Fritz Grunewald, Daniel Segal, and Geoffrey Smith, Subgroups of Finite Index in Nilpotent Groups, Invent. Math. 93 (1988), no. 1, 185–223. MathSciNet
[4] Andrei Jaikin-Zapirain, On the Number of Conjugacy Classes of Finite p-Groups, J. London Math. Soc. (2) 68 (2003), no. 3, 699–711. MathSciNet
[5] Alexander Moretó, Complex Group Algebras of Finite Groups: Brauer's Problem 1, Adv. Math. 208 (2007), no. 1, 236–248. MathSciNet