Séminaire Lotharingien de Combinatoire, B56a (2006), 32 pp.

Thomas W. Müller

Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks

Abstract. We survey a number of powerful recent results concerning diophantine and asymptotic properties of (ordinary) characters of symmetric groups. Apart from their intrinsic interest, these results are motivated by a connection with subgroup growth theory and the theory of random walks. As applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, as well as a finiteness result for the number of Fuchsian presentations of such a group, the latter result solving a long-standing problem of Roger Lyndon's. We also sketch the proof of a well-known conjecture of Roichman's concerning the mixing time of random walks on finite symmetric groups, and of a result describing the parity of the subgroup numbers for a substantial class of one-relator groups.

Received: July 24, 2006. Revised: August 23, 2006. Accepted: August 23, 2006.

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