A note on surfaces bounding hyperbolic 3-manifolds (Q701810)

The author discusses the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold. He presents the following two fairly symmetric examples related to the Hurwitz action and to the \(\mathbb A_5\)-action. Let \(F\) be a hyperbolic surface of genus \(g\) which admits a group \(G\) of orientation preserving isometries of maximal possible order \(84(g-1)\) (``Hurwitz action''). Then \(F\) occurs as a boundary component of a compact hyperbolic 3-manifold with totally geodesic boundary. The hyperbolic surface \(F\) of genus 4 associated to the small stellated dodecahedron is the totally geodesic boundary of the compact hyperbolic 3-manifold \(M\) obtained from the truncated regular hyperbolic icosahedron with dihedral angles \(2 \pi / 5\) by identifying opposite faces after a twist of \(2 \pi / 6\). The isometric \(\mathbb A_5\)-action on \(F\) extends to the isometric \(\mathbb A_5\)-action on \(M\), where \(\mathbb A_5\) denotes the dodecahedral group of order 60 (the orientation preserving symmetry group of both the small stellated dodecahedron and the icosahedron).