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Andrew Putman

  1. Representation stability, congruence subgroups, and mapping class groups.

    Wed, 01/25/2012 - 15:01 — SomNambul
    Authors: Andrew Putman
    Subjects: Algebraic Topology
    Abstract

    The homology groups of many natural sequences of groups
    $\{G_n\}_{n=1}^{\infty}$ (e.g.\ general linear groups, mapping class groups,
    etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known
    machine for proving such results that goes back to early work of Quillen.
    Church and Farb discovered that many sequences of groups whose homology groups
    do not stabilize in the classical sense actually stabilize in some sense as
    representations. They called this phenomena {\em representation stability}.

  2. Abelian quotients of subgroups of the mapping class group and higher Prym representations.

    Wed, 06/15/2011 - 15:01 — SomNambul
    Authors: Andrew Putman, Ben Wieland
    Subjects: Geometric Topology
    Abstract

    A well-known conjecture asserts that the mapping class group of a surface
    (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the
    genus of the surface is large. We prove that if this conjecture holds for some
    genus, then it also holds for all larger genera. We also prove that if there is
    a counterexample to this conjecture, then there must be a counterexample of a
    particularly simple form. We prove these results by relating the conjecture to
    a family of linear representations of the mapping class group that we call the
    higher Prym representations.

  3. A Birman exact sequence for Aut(F_n).

    Thu, 04/14/2011 - 15:01 — SomNambul
    Authors: Matthew B. Day, Andrew Putman
    Subjects: Geometric Topology
    Abstract

    The Birman exact sequence describes the effect on the mapping class group of
    a surface with boundary of gluing discs to the boundary components. We
    construct an analogous exact sequence for the automorphism group of a free
    group. For the mapping class group, the kernel of the Birman exact sequence is
    a surface braid group. We prove that in the context of the automorphism group
    of a free group, the natural kernel is finitely generated.

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