Quasi-isometries of graph manifolds do not preserve non-positive curvature
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2014
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ndltd-OhioLink-oai-etd.ohiolink.edu-osu14058946402021-08-03T06:26:00Z Quasi-isometries of graph manifolds do not preserve non-positive curvature Nicol, Andrew Mathematics graph manifolds geometric group theory quasi-isometries non-positive curvature bounded cohomology relative hyperbolicity isolated flats 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. 2014-10-15 English text The Ohio State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640 http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
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NDLTD |
| language |
English |
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NDLTD |
| topic |
Mathematics graph manifolds geometric group theory quasi-isometries non-positive curvature bounded cohomology relative hyperbolicity isolated flats |
| spellingShingle |
Mathematics graph manifolds geometric group theory quasi-isometries non-positive curvature bounded cohomology relative hyperbolicity isolated flats Nicol, Andrew Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| author |
Nicol, Andrew |
| author_facet |
Nicol, Andrew |
| author_sort |
Nicol, Andrew |
| title |
Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| title_short |
Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| title_full |
Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| title_fullStr |
Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| title_full_unstemmed |
Quasi-isometries of graph manifolds do not preserve non-positive curvature |
| title_sort |
quasi-isometries of graph manifolds do not preserve non-positive curvature |
| publisher |
The Ohio State University / OhioLINK |
| publishDate |
2014 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640 |
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AT nicolandrew quasiisometriesofgraphmanifoldsdonotpreservenonpositivecurvature |
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1719436808537767936 |