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Character Theory of Symmetric Groups and Subgroup Growth of Surface Groups
Author(s) -
Müller Thomas W.,
Puchta Jan-Christoph
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003599
Subject(s) - mathematics , character (mathematics) , rank (graph theory) , group (periodic table) , combinatorics , symmetric group , characteristic subgroup , fitting subgroup , maximal subgroup , normal subgroup , commutator subgroup , index of a subgroup , pure mathematics , property (philosophy) , p group , geometry , physics , philosophy , epistemology , quantum mechanics
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/(χ λ (1)) s taken over all partitions λ of n with fixed s ⩾ 1, which is also established.