We use H C Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group to construct an explicit lower bound for the volume of a hyperbolic n-orbifold.
57M50, 57N16; 20H10, 22E40
IntroductionLet H n denote hyperbolic n-space, the unique simply connected n-dimensional Riemannian manifold of constant sectional curvature 1. A hyperbolic n-orbifold Q is a quotient H n = , where represents a discrete group of orientation-preserving isometries. A hyperbolic n-orbifold is a manifold when contain no elements of finite order. Martin [17] constructed a lower bound for r n , the largest number such that every hyperbolic n-manifold contains a round ball of that radius; see also Friedland and Hersonsky [8]. From this one can compute, in each dimension, an explicit lower bound for the volume of a hyperbolic n-manifold.The purpose of this paper is to give an explicit lower bound for the volume of a hyperbolic n-orbifold, again depending only on dimension. The result of this article is more general than what was achieved in the prequel [1]. Our work also significantly improves upon the volume bounds of [1; 17], even though we consider a larger category of orbit spaces.We define a Riemannian submersion W SO o .n; 1/= ! H n = , where the connected component SO o .n; 1/ of the identity in the Lie group O.n; 1/ is isomorphic to the full group of orientation-preserving isometries of H n . The study of the volume of a hyperbolic orbifold is thereby reduced to the study of the covolume of a lattice in a Lie group.Wang [23] showed that the covolume of a lattice in a semisimple Lie group that contains no compact factor can be bounded below by the volume of ball with a radius that depends only on the group itself. We estimate the sectional curvature of SO o .n; 1/ and apply a comparison theorem due to Gunther (see eg Gallot, Hulin and Lafontaine [10]), to produce a lower bound for VolOESO o .n; 1/= . The following theorem gives our main result.
We derive an explicit lower bound for the volume of a hyperbolic orbifold, dependent on the dimension and the maximal order of torsion in the orbifold's fundamental group.
For a closed, strictly convex projective manifold of dimension n ≥ 3 that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson-Courtois-Gallot's entropy rigidity result to Hilbert geometries.
The area of a convex projective surface of genus [Formula: see text] is at least [Formula: see text] where [Formula: see text] is the vector of triangle invariants of Bonahon–Dreyer and [Formula: see text] are the Fock–Goncharov triple ratios.
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