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Scott H. Murray

  1. Constructive homomorphisms for classical groups.

    Fri, 09/10/2010 - 15:01 — SomNambul
    Authors: Scott H. Murray, Colva M. Roney-Dougal
    Subjects: Group Theory
    Abstract

    Let Omega be a quasisimple classical group in its natural representation over
    a finite vector space V, and let Delta be its normaliser in the general linear
    group. We construct the projection from Delta to Delta/Omega and provide fast,
    polynomial-time algorithms for computing the image of an element. Given a
    discrete logarithm oracle, we also represent Delta/Omega as a group with at
    most 3 generators and 6 relations. We then compute canonical representatives
    for the cosets of Omega.

  2. Fundamental domains for congruence subgroups of SL2 in positive characteristic.

    Wed, 09/02/2009 - 12:02 — SomNambul
    Authors: Lisa Carbone, Leigh Cobbs, Scott H. Murray
    Subjects: Group Theory
    Abstract

    Morgenstern ([M]) claimed to have constructed fundamental domains for
    congruence subgroups of the lattice group Gamma=PGL_2(F_q[t]), and subgraphs
    providing the first known examples of linear families of bounded concentrators.
    His method was to construct the fundamental domain for a congruence subgroup as
    a `ramified covering' of the fundamental domain for Gamma on the Bruhat-Tits
    tree X of G=PGL_2(F_q((t^-1))). We prove that Morgenstern's constructions do
    not yield the desired ramified coverings, and in particular yield graphs that
    are not connected in characteristic 2.

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