Group actions, geodesic loops, and symmetries of compact hyperbolic 3-manifolds

Compact hyperbolic 3-manifolds are used in cosmological models. Their topology is characterized by their homotopy group $\pi_1(M)$ whose elements multiply by path concatenation. The universal covering of the compact manifold $M$ is the hyperbolic space $H^3$ or the hyperbolic ball $B^3$. They share with $M$ a Riemannian metric of constant negative curvature and allow for the isometric action of the group $Sl(2,C)$. The homotopy group $\pi_1(M)$ acts as a uniform lattice $\Gamma(M)$ on $B^3$ and tesselates it by copies of $M$. Its elements $g$ produce preimage and image points for geodesic sections on $B^3$ which by self-intersection form geodesic loops on $M$. For any fixed hyperbolic $g \in \Gamma$ we construct a continuous commutative two-parameter normalizer $N_g <Sl(2,C)$ and its orbit surfaces on $B^3$. The orbit surfaces classify sets of geodesic loops of equal length. We give general expressions for the length of geodesic loops and for the defect angle at the self-intersection on $M$ in terms of the group parameters of $g$ and orbit parameters on $B^3$. Geodesic loops of minimal length, given from the character $\chi(g)$, belong to a single orbit. These and only these minimal geodesic loops have vanishing defect angle and hence are smooth everywhere. The role of symmetries is illuminated by the example of the dodecahedral hyperbolic Weber-Seifert manifold $M$. $\Gamma(M)$ is normal in the hyperbolic Coxeter group with Coxeter diagram ${\bf \circ \frac{5}{}\circ\frac{3}{}\circ\frac{5}{}\circ}$. This leads to symmetry relations between geodesic loops.