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Register a new userOn volumes of quaternionic hyperbolic n-orbifolds
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By use of H. C. Wangs bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension.
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In this paper, we obtain analogues of Jorgensens inequality for non-elementary groups of isometries of quaternionic hyperbolic $n$-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker [7] As applications, we use the quaternionic version of J{o}rgensens inequalities to construct embedded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show that these canonical collars are disjoint from each other. Our results give some improvement over earlier results of Markham and Parker and answer an open question posed in [16].
We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixture of finite, ideal, and truncated (i.e. ultra-ideal) vertices. A consequence of this decomposition is that the volumes of spherical tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces S_g x I with genus g at least 2. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for links in S_g x I, ranging over all g at least 2, is a dense subset of the interval [0, 2v_{oct}], where v_{oct}, approximately 3.66386, is the volume of the regular ideal octahedron.
Jorgensens inequality gives a necessary condition for a non-elementary two generator group of isometries of real hyperbolic 2-space to be discrete. We give analogues of Jorgensens inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic.
We show that Mirzakhanis curve counting theorem also holds if we replace surfaces by orbifolds.
In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of $SL_{2}left(mathbb{Z}right)$ and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.
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