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Quasi-isometries of graph manifolds do not preserve non-positive curvature

Nicol, Andrew
http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640

Abstract Details

2014, Doctor of Philosophy, Ohio State University, Mathematics.
Continuing the work of Frigerio, Lafont, and Sisto, we recall the definition of high dimensional graph manifolds. They are compact, smooth manifolds which decompose into finitely many pieces, each of which is a hyperbolic, non-compact, finite volume manifold of some dimension with toric cusps which has been truncated at the cusps and crossed with an appropriate dimensional torus. This class of manifolds is a generalization of two of the geometries described in Thurston's geometrization conjecture: three dimensional hyperbolic space and the product of two dimensional hyperbolic space with R. In their monograph, Frigerio, Lafont, and Sisto describe various rigidity results and prove the existence of infinitely many graph manifolds not supporting a locally CAT(0) metric. Their proof relies on the existence of a cochain having infinite order. They leave open the question of whether or not there exists pairs of graph manifolds with quasi-isometric fundamental group but where one supports a locally CAT(0) metric while the other cannot. Using a number of facts about bounded cohomology and relative hyperbolicity, I extend their result, showing that there exists a cochain that is not only of infinite order but is also bounded. I also show that this is sufficient to construct two graph manifolds with the desired properties.
Jean-Fracois Lafont (Advisor)
Michael Davis (Committee Member)
Nathan Broaddus (Committee Member)
65 p.
Mathematics
graph manifolds; geometric group theory; quasi-isometries; non-positive curvature; bounded cohomology; relative hyperbolicity; isolated flats

Recommended Citations

Citations

  • Nicol, A. (2014). Quasi-isometries of graph manifolds do not preserve non-positive curvature [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640

    APA Style (7th edition)

  • Nicol, Andrew. Quasi-isometries of graph manifolds do not preserve non-positive curvature. 2014. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640.

    MLA Style (8th edition)

  • Nicol, Andrew. "Quasi-isometries of graph manifolds do not preserve non-positive curvature." Doctoral dissertation, Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640

    Chicago Manual of Style (17th edition)

osu1405894640
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This open access ETD is published by The Ohio State University and OhioLINK.