Constructing hyperbolic manifolds (Q2702047)

The Coxeter group \(\Gamma\), generated by reflections in the side hyperplanes of the Coxeter 4-simplex with symbol \(\circ\diagrBar\circ\multimap\circ\diagrBar\circ\), acts on the hyperbolic 4-space \(H^4\). The vertex stabilizer with \(\circ\diagrbar\circ\diagrBar\circ\) is a spherical Coxeter group, as symmetries of the regular 120-cell of \(S^3\). The famous Davis 4-manifold \(H^4/K\) can be derived by the torsion-free kernel \(K\) of an epimorphism \(\Gamma\to G\). The authors analyse the more general situation where \(\Gamma\) is a Coxeter group acting on \(H^n\), and \(G\) is a certain classical group. The basic methods are taken from \textit{E. B. Vinberg} [e.g. Discrete groups generated by reflections in Lobacevskii spaces, Mat. Sb., N. Ser. 72(114), 471-488, 73(115), 303 (1967; Zbl 0166.16303)]. As a special case, the authors obtain further information on the quotient manifold \(H^n/K\) as above.NEWLINENEWLINENEWLINEThe reviewer remarks that the homomorphism \(\Gamma\to G\) is not a necessary assumption for constructing a torsion-free \(K<\Gamma\) and so the manifold \(H^n/K\). It would be interesting to construct a manifold different from the one of Davis, analogously as we know other dodecahedron spaces different from the Weber-Seifert one.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].