• About
    • Login
    View Item 
    •   Institutional Repository Home
    • Electronic Theses and Dissertations
    • Electronic Theses and Dissertations
    • View Item
    •   Institutional Repository Home
    • Electronic Theses and Dissertations
    • Electronic Theses and Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Browse

    All of Institutional RepositoryCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects

    My Account

    LoginRegister

    Subgroup Distortion in Metabelian and Free Nilpotent Groups

    Davis, Tara Colleen
    : https://etd.library.vanderbilt.edu/etd-03222011-131219
    http://hdl.handle.net/1803/11027
    : 2011-04-02

    Abstract

    The main result of this dissertation sheds light on subgroup distortion in metabelian and free nilpotent groups. A subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F. Also, the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian are studied. It is shown that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k , there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated) subgroups. Following the work done by Olshanskii for groups, it is also described, for a given semigroup S, which functions l : S → N can be realized up to equivalence as length functions g ↦→ |g|H by embedding S into a finitely generated semigroup H. Following the work done by Olshanskii and Sapir, a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup is provided. This classification for groups has connections with another function of interest in geometric group theory: the relative growth function. There are connections between the relative growth of cyclic subgroups, and the corresponding distortion function of the embedding. In particular, when the distortion is non-recursive, the relative growth is at least almost quadratic. On the other hand, there exists a cyclic subgroup of a two generated group such that the distortion function associated to the embedding is not bounded above by any recursive function, and yet the relative growth is o(r^2).
    Show full item record

    Files in this item

    Icon
    Name:
    davis_t2.pdf
    Size:
    391.2Kb
    Format:
    PDF
    View/Open

    This item appears in the following collection(s):

    • Electronic Theses and Dissertations

    Connect with Vanderbilt Libraries

    Your Vanderbilt

    • Alumni
    • Current Students
    • Faculty & Staff
    • International Students
    • Media
    • Parents & Family
    • Prospective Students
    • Researchers
    • Sports Fans
    • Visitors & Neighbors

    Support the Jean and Alexander Heard Libraries

    Support the Library...Give Now

    Gifts to the Libraries support the learning and research needs of the entire Vanderbilt community. Learn more about giving to the Libraries.

    Become a Friend of the Libraries

    Quick Links

    • Hours
    • About
    • Employment
    • Staff Directory
    • Accessibility Services
    • Contact
    • Vanderbilt Home
    • Privacy Policy