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Topological Rigidity of Hyperbolic Manifolds with Piecewise Totally Geodesic Boundary

Chaparro Sumalave, Gustavo
http://rave.ohiolink.edu/etdc/view?acc_num=osu1689553090106198

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
The relative version of the Borel Conjecture regarding topological rigidity of manifolds with boundary posits that any homotopy equivalence between compact aspherical manifolds that restricts to a homeomorphism on the boundary is homotopic (relative to the boundary) to a homeomorphism. Frigerio shows that any isomorphism of fundamental groups between finite volume \( n \)-dimensional (\( n\ge 4 \)) hyperbolic manifolds with \textit{totally geodesic} boundary is induced by an isometry, which, in particular, produces a homeomorphism between the spaces. As a step toward geometric rigidity, in this thesis we prove that there is topological rigidity for high dimensional hyperbolic manifolds whose boundary is \textit{piecewise totally geodesic}. More specifically, let \( \mathcal M \) be the collection of compact hyperbolic manifolds \( M \) with non-empty boundary such that: the dimension of \( M \) is at least \( 7 \); the boundary \( \partial M \) is the union \( N_1\cup_W N_2 \) of two totally geodesic submanifolds \( N_1 \) and \( N_2 \) of \( M \) that intersect along a codimension 2 submanifold \( W \); the dihedral angle \( \theta_M \) formed at every point in \( W \) is constant. We consider two different cases depending on the direction of the bend along \( W \): \begin{itemize} \item Inward angle manifolds: \( \pi/2 < \theta_M\le \pi \); the injectivity radius of \( W \subset \partial M \) is large, and geodesic arcs with endpoints on \( \partial M \) are large. \item Outward angle manifolds: \( \pi \le \theta_M < 3\pi/2 \) and the injectivity radius of \( W\subset \partial M \) is large. \end{itemize} For any \( M_1,M_2\in \mathcal M \) in either of the two cases, we show that \( M_1 \) and \( M_2 \) are homeomorphic if and only if \( \pi_1(M_1) \) and \( \pi_1(M_2) \) are isomorphic. The proof relies on a rigidity result by Lafont and Tshishiku for hyperbolic groups whose boundary at infinity is a Sierpinski curve. We also discuss how to explicitly construct infinitely many examples of inward and outward angle manifolds using arithmetic hyperbolic manifolds.
Jean-Francois Lafont (Advisor)
Jingyin Huang (Committee Member)
Nathan Broaddus (Committee Member)
Robert de Jong (Committee Member)
63 p.
Mathematics
topological rigidity; hyperbolic manifolds; totally geodesic boundary; boundary at infinity

Recommended Citations

Citations

  • Chaparro Sumalave, G. (2023). Topological Rigidity of Hyperbolic Manifolds with Piecewise Totally Geodesic Boundary [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689553090106198

    APA Style (7th edition)

  • Chaparro Sumalave, Gustavo. Topological Rigidity of Hyperbolic Manifolds with Piecewise Totally Geodesic Boundary. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1689553090106198.

    MLA Style (8th edition)

  • Chaparro Sumalave, Gustavo. "Topological Rigidity of Hyperbolic Manifolds with Piecewise Totally Geodesic Boundary." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1689553090106198

    Chicago Manual of Style (17th edition)

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