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Quaternions and some global properties of hyperbolic 5-manifolds. (English) Zbl 1054.57019

Let \(M\) be an oriented complete hyperbolic \(5\)-manifold of finite volume and \(i_p(M)\) the injectivity radius of \(M\) at \(p\). By the Margulis Lemma for discrete groups of hyperbolic isometries (see W. Ballmann, M. Gromov and V. Schroeder [Manifolds of Nonpositive Curvature, (Section 9–10, Progress in Mathematics, 61. Boston-Basel-Stuttgart: Birkhäuser), (1985; Zbl 0591.53001)]; W. P. Thurston [Three-dimensional geometry and topology, Vol. 1, (Princeton University Press), (1997; Zbl 0873.57001)]; J.G. Ratcliffe, Foundations of hyperbolic manifolds, (Springer-Verlag, Berlin), (1994; Zbl 0809.51001)]) there is a universal positive constant \(\varepsilon=\varepsilon_5\) such that there is a thick and thin decomposition \(M=M_{\leq\varepsilon}\cup M_{>\varepsilon}\), where the thick part \(M_{>\varepsilon}=\{p\in M\;| \;i_p(M)>\varepsilon/2\}\) of \(M\) is compact, and the thin part \(M_{\leq\varepsilon}=\{p\in M\;| \;i_p(M)\leq\varepsilon/2\}\) consists of connected components. Taking up an idea of R. Meyerhoff [Can. J. Math. 39, 1038–1056 (1987; Zbl 0694.57005)]) the author proves that for \(\varepsilon\leq\sqrt{3}/9\pi\) there is such a decomposition of \(M\) and the thin part \(M_{\leq\varepsilon}\) is a finite disjoint union of canonical cusps and tubes around simple closed geodesics of length \(l\leq\sqrt{3}/8\pi\) of radius \(r\) given by \(\text{cosh}(2r)=\frac{1-3k}{k}\), where \(k=2\pi l/\sqrt{3}\). The tubes around distinct closed geodesics of length \(\leq\sqrt{3}/9\pi\) are pairwise disjoint and in the non-compact case, they are also distinct from the canonical cusps associated to parabolic elements in the fundamental group of \(M\). Explicit values for the constant \(\varepsilon\) are also known for the dimension two [P. Buser, Geometry and spectra of compact Riemann surfaces, (Chapter 4, Birkhäuser), (1992; Zbl 0770.53001)] and three [R. Meyerhoff, loc. cit.] For dimension four, there are partial results (see the author, [Ann. Global Anal. Geom. 13, No. 4, 377–392 (1995; Zbl 0874.52002)]). As applications, the author derives new universal lower bounds for the volume \(vol_5(M)\) of \(M\) and, in the case when \(M\) is compact, new estimates relating the injectivity radius \(i(M)\) of \(M\) and the diameter of \(M\) with \(vol_5(M)\). For example, one proves \(i(M)\geq const(vol_5(M))^{-1}\), which improves results of P. Buser [Geometry of the Laplace operator, Honolulu/Hawaii 1979, Proc. Symp. Pure Math., Vol. 36, 29–77 (1980; Zbl 0432.58024)] and A. Reznikov [Topology 34, No.2, 477–479 (1995; Zbl 1008.53041)].

MSC:

57M50 General geometric structures on low-dimensional manifolds
22E40 Discrete subgroups of Lie groups
57N16 Geometric structures on manifolds of high or arbitrary dimension
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