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Register a new userJorgensens Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space
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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].
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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.
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.
In this paper, we give an analogue of Jorgensens inequality for non-elementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jorgensens inequality in 2-dimensional Mobius group of the above case.
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on these spaces which are of their independent interests. To this end, we need to identify and introduce the ``Quaternionic Gellers operators and ``Octonionic Gellers operators which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernel estimates and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient for us to establish, as a byproduct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works, completes our study of the factorization theorems, higher order Poincare-Sobolev, Hardy-Sobolev-Mazya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.
A 3D rep-tile is a compact 3-manifold $X$ in $mathbb{R}^3$ that can be decomposed into finitely many pieces, each of which are similar to $X$, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected graph in $S^3$.
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