CHARACTER THEORY OF SYMMETRIC GROUPS AND SUBGROUP GROWTH OF SURFACE GROUPS

Abstract

Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a ‘large’ surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum $\Sigma 1/(\chi_\lambda(1))^s$ taken over all partitions $\lambda$ of $n$ with fixed $s \ge 1$ , which is also established.