Coarse-Geometric Perspective on Negatively Curved Manifolds and Groups

Alex Furman ()
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Alex Furman: University of Illinois at Chicago, Department of Mathematics

A chapter in Rigidity in Dynamics and Geometry, 2002, pp 149-166 from Springer

Abstract: Abstract Let (X, g) be a compact negatively curved Riemannian manifold with fundamental group Γ. Restricting the lifted metric on the universal cover $$\left( {\tilde X,\tilde g} \right)$$ of (X, g) to a Γ-orbit, Γx, one obtains a left invariant metric d g, x on Γ, which is well defined up to a bounded amount, depending on the choice of the orbit Γx. Motivated by this geometric example, we study classes [d]of general left-invariant metrics d on general Gromov hyperbolic groups Γ, where [d 1]= [d 2]if d 1 — d 2 is bounded. It turns out that many of the geometric objects associated with (X, g) — such as marked length spectrum, crossratio, Bowen-Margulis measure — can be defined in the general coarse-geometric setting. The main result of the paper is a characterization of the compact negatively curved locally symmetric spaces within this coarse-geometric setting.

Keywords: Symmetric Space; Fundamental Group; Compact Group; Hyperbolic Group; Geodesic Flow (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-04743-9_7

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DOI: 10.1007/978-3-662-04743-9_7

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