  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.
