Spherical Coxeter groups and hyperelliptic 3-manifolds. (English. Russian original) Zbl 0948.57010

Vesnin, A. Yu.; Mednykh, A. D.

Math. Notes 66, No. 2, 135-138 (1999); translation from Mat. Zametki 66, No. 2, 173-177 (1999).

A 3-manifold \(M\) is called hyperelliptic if there exists an isometric involution \(\tau\) such that the quotient space \(M/\langle\tau\rangle\) is homeomorphic to the 3-sphere \(S^3\). Let \(X\) be one of the 3-dimensional geometries \(H^3\), \(E^3\), \(S^3\), \(H^2 \times E^1\), or \(S^2 \times E^1\), and let \(P\) be a Coxeter polyhedron (that is, each dihedral angle of \(P\) is of the form \(\pi/n\) for some integer \(n\)) in \(X\). Let \(\Delta(P)\) denote the group of isometries of \(X\) generated by reflections in the faces of \(P\). Under a condition relating \(P\) to some Coxeter polyhedron in \(S^3\), the authors show that \(\Delta(P)\) contains a torsion free, finite index subgroup \(G\) such that \(X^3/G\) is a hyperelliptic manifold. This result generalizes earlier work of the second author [Ann. Global Anal. Geom. 8, No. 1, 13-19 (1990; Zbl 0712.57007)].

Reviewer: J.Hebda (St.Louis)

MSC:

57M50	General geometric structures on low-dimensional manifolds
57N10	Topology of general \(3\)-manifolds (MSC2010)
20F55	Reflection and Coxeter groups (group-theoretic aspects)

Keywords:

spherical Coxeter polyhedron; reflection group; hyperelliptic manifold; Coxeter group; skeleton of a Coxeter polyhedron; Thurston geometries

Citations:

Zbl 0712.57007

References:

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