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

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.

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Nicol, A. (2014). Quasi-isometries of graph manifolds do not preserve non-positive curvature. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Nicol, Andrew. "Quasi-isometries of graph manifolds do not preserve non-positive curvature." Electronic Thesis or Dissertation. Ohio State University, 2014. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Nicol, Andrew "Quasi-isometries of graph manifolds do not preserve non-positive curvature." Electronic Thesis or Dissertation. Ohio State University, 2014. https://etd.ohiolink.edu/

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