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Probabilistic Group Theory and Fuchsian Groups

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Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Part of the book series: Progress in Mathematics ((PM,volume 248))

Abstract

In this paper we survey recent advances in probabilistic group theory and related topics, with particular emphasis on Fuchsian groups. Combining a character-theoretic approach with probabilistic ideas we study the spaces of homomorphisms from Fuchsian groups to symmetric groups and to finite simple groups, and obtain a whole range of applications.

In particular we show that a random homomorphism from a Fuchsian group to the alternating group An is surjective with probability tending to 1 as n → ∞, and this implies Higman’s conjecture that every Fuchsian group surjects to all large enough alternating groups. We also show that a random homomorphism from a Fuchsian group of genus ≥ 2 to any finite simple group G is surjective with probability tending to 1 as |G| → ∞. These results can be viewed as far-reaching extensions of Dixon’s conjecture, establishing a similar result for free groups.

Other applications concern counting branched coverings of Riemann surfaces, studying the subgroup growth of Fuchsian groups, and computing the dimensions of representation varieties of Fuchsian groups over fields of arbitrary characteristic.

A main tool in our proofs are results of independent interest which we obtain on the representation growth of symmetric groups and groups of Lie type. These character-theoretic results also enable us to analyze random walks in symmetric groups and in finite simple groups, with certain conjugacy classes as generating sets, and to determine the precise mixing time.

Most of the results outlined here were proved in recent joint works by Liebeck and myself, but we shall also describe related results by other authors.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Aner Shalev

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  1. Aner Shalev
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Editors and Affiliations

  1. Institut de Mathématiques, EPFL, 1015, Lausanne, Switzerland

    Laurent Bartholdi

  2. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 108, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  3. Section de mathématiques, Université de Genève, 1211, Geneva 4, Switzerland

    Tatiana Smirnova-Nagnibeda

  4. Institut de Mathématiques, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Andrzej Zuk

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To Slava Grigorchuk on his 50th birthday

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Shalev, A. (2005). Probabilistic Group Theory and Fuchsian Groups. In: Bartholdi, L., Ceccherini-Silberstein, T., Smirnova-Nagnibeda, T., Zuk, A. (eds) Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol 248. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7447-0_9

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