Character Theory of Symmetric Groups and Subgroup Growth of Surface Groups

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 Σ1/(χλ(1))s taken over all partitions λ of n with fixed s ⩾ 1, which is also established.

[1]  M. Hall,et al.  On Schreier systems in free groups , 1948 .

[2]  M. Hall.  Subgroups of Finite Index in Free Groups , 1949, Canadian Journal of Mathematics.

[3]  M. Hall.  Coset representations in free groups , 1949 .

[4]  J. S. Frame,et al.  The Hook Graphs of the Symmetric Group , 1954, Canadian Journal of Mathematics.

[5]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[6]  E. M. Wright.  Asymptotic Relations Between Enumerative Functions in Graph Theory , 1970 .

[7]  E. Bender.  An asymptotic expansion for the coefficients of some formal power series , 1975 .

[8]  Asymptotic formulas related to free products of cyclic groups , 1976 .

[9]  Stephen J. Pride,et al.  Groups with two More Generators Than Relators , 1978 .

[10]  Alan Williamson,et al.  The Probability of Generating the Symmetric Group , 1978 .

[11]  Amitai Regev,et al.  Asymptotic values for degrees associated with strips of young diagrams , 1981 .

[12]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[13]  Irving Reiner,et al.  Methods of Representation Theory , 1981 .

[14]  A. Mednykh.  On the number of subgroups in the fundamental group of a closed surface , 1988 .

[15]  Marcus du Sautoy,et al.  Finitely generated groups, $p$-adic analytic groups, and Poincaré series , 1990 .

[16]  Adalbert Kerber,et al.  Algebraic combinatorics via finite group actions , 1991 .

[17]  Die Wurzelanzahlfunktion in symmetrischen Gruppen , 1991 .

[18]  Combinatorial Aspects of Finitely Generated Virtually Free Groups , 1991 .

[19]  Counting free subgroups of finite index , 1992 .

[20]  Finitely generated groups, $p$-adic analytic groups and Poincaré series , 1993 .

[21]  Avinoam Mann,et al.  Finitely generated groups of polynomial subgroup growth , 1993 .

[22]  A. Lubotzky.  Subgroup growth and congruence subgroups , 1995 .

[23]  Christian Krattenthaler,et al.  Bijective proofs of the hook formulas for the number of standard Young tableaux, ordinary and shifted , 1995, Electron. J. Comb..

[24]  T. Müller.  Subgroup growth of free products , 1996 .

[25]  Combinatorial Classification of Finitely Generated Virtually Free Groups , 1997 .

[26]  Andreas W. M. Dress,et al.  Decomposable Functors and the Exponential Principle , 1997 .

[27]  Igor Pak,et al.  A direct bijective proof of the hook-length formula , 1997, Discret. Math. Theor. Comput. Sci..

[28]  Thomas Müller.  Enumerating Representations in Finite Wreath Products , 2000 .

[29]  Analytic properties of zeta functions and subgroup growth , 2000, math/0011267.

[30]  Igor Pak,et al.  Hook length formula and geometric combinatorics. , 2001 .

[31]  T. Müller.  Modular Subgroup Arithmetic and a Theorem of Philip Hall , 2002 .

[32]  T. Müller,et al.  Parity patterns in one-relator groups , 2003 .

[33]  Modular subgroup arithmetic , 2003 .

[34]  T. Müller.  Modular subgroup arithmetic in free products , 2003 .