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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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