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# Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds

Published:2003-10-01
Printed: Oct 2003
• Ruth Kellerhals
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## Abstract

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.
 MSC Classifications: 53C22 - Geodesics [See also 58E10] 53C25 - Special Riemannian manifolds (Einstein, Sasakian, etc.) 57N16 - Geometric structures on manifolds [See also 57M50] 57S30 - Discontinuous groups of transformations 51N30 - Geometry of classical groups [See also 20Gxx, 14L35] 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]