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    Automorphisms of nonpositively curved cube complexes, right-angled Artin groups and homology

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    Author
    Bregman, Corey Joseph
    Date
    2017-04-20
    Advisor
    Putman, Andrew; Wolf, Michael
    Degree
    Doctor of Philosophy
    Abstract
    Recently, the geometry of CAT(0) cube complexes featured prominently in Agol’s resolution of two longstanding conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. A key step of the proof was to show that every hyperbolic 3-manifold group is virtually special, i.e. virtually the fundamental group of a special nonpositively curved (NPC) cube complex. In this thesis, we study algebraic properties of special groups as they relate to the geometry of special cube complexes. In the first part of the thesis, we introduce a positive integer-valued invariant of special cube complexes called the genus, and show that having genus one is equivalent to having free abelian fundamental group. As a corollary, we obtain a new proof of the fact that every special group is either abelian or surjects onto a non-abelian free group. In the second part of the thesis, we turn our attention to automorphisms of NPC cube complexes. We give a criterion on a special cube complex which implies that any automorphism acts non-trivially on first homology, and show that a non- trivial action on homology can always be achieved by passing to covers. We then apply the criterion to provide a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.
    Keyword
    Geometric group theory; CAT(0) geometry; low-dimensional topology
    Citation
    Bregman, Corey Joseph. "Automorphisms of nonpositively curved cube complexes, right-angled Artin groups and homology." (2017) Diss., Rice University. https://hdl.handle.net/1911/96119.
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    Home | FAQ | Contact Us | Privacy Notice | Accessibility Statement
    Managed by the Digital Scholarship Services at Fondren Library, Rice University
    Physical Address: 6100 Main Street, Houston, Texas 77005
    Mailing Address: MS-44, P.O.BOX 1892, Houston, Texas 77251-1892
    Site Map