Coxeter groups and branched coverings of lens spaces (Q2770398)

The authors consider 3-manifolds \(M^3\) which are quotient spaces \(M^3=X^3/\Gamma\), where \(X^3\) is one of the 3-dimensional geometries \(E^3\), \(H^3\), \(S^3\), \(H^2\times E^1\) or \(S^2\times E^1\), and \(\Gamma\) is a discrete group of isometries acting on \(X^3\) without fixed points. \(M^3\) is called \(g\)-hyperelliptic if there exists an isometric involution \(\tau\) on \(M^3\) such that \(N^3=M^3/\langle\tau\rangle\) is a 3-manifold of Heegaard genus \(g\). Hence, 0-hyperelliptic manifolds are hyperelliptic manifolds in the usual sense, and they have been intensively studied. In this paper the case \(g=1\) is examined, where obviously \(N^3\) is a lens space. The authors give a constructive description of 1-hyperelliptic manifolds in terms of finite index subgroups of Coxeter groups acting on 3-dimensional spaces.