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Title: Volume and rigidity of hyperbolic polyhedral 3-manifolds: VOLUME AND RIGIDITY OF HYPERBOLIC POLYHEDRAL 3-MANIFOLDS
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Journal Name:
Journal of Topology
Page Range or eLocation-ID:
1 to 29
Oxford University Press (OUP)
Sponsoring Org:
National Science Foundation
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  1. Abstract

    We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary.

  2. A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$ , and complements well-known results on Euclidean and spherical rank rigidity.