Interesting classes of incomplete hyperbolic manifolds with torsion?

Question

I am wondering if there are any interesting classes of incomplete hyperbolic manifolds? In particular, are there any interesting families with torsion (i.e. a cyclic subgroup in $\pi_1$)? There are orbifolds with torsion, but that's not what I'm looking for. I tried adapting the construction for lens spaces to the hyperbolic case and I was suggested to take copies of orbifolds and remove a point, but neither seem to work.

Answer 1

Here is one class of interesting examples (they are all unorientable). Start with the "central symmetry" of the open unit ball ${\mathbb B}^3$: $$ s: {\mathbf x}\mapsto -{\mathbf x}. $$ The map $s$ has single fixed point, the origin $0$. I will equip ${\mathbb B}^3$ with the standard hyperbolic metric. The quotient $({\mathbb B}^3- \{0\})/s$ is an incomplete hyperbolic manifold whose fundamental group is finite cyclic, of order 2.

Now, imagine having a more complicated (infinite) discrete subgroup $\Gamma$ of isometries of the hyperbolic 3-space containing conjugates of $s$ and no other nontrivial finite order elements. Then, removing from ${\mathbb B}^3$ the (discrete) set of fixed points of order 2 elements in $\Gamma$ we obtain an incomplete Riemannian manifold $X$; the quotient $X/\Gamma$ is an incomplete hyperbolic manifold $M$ whose fundamental group contains infinitely many elements of order $2$. You can think of $M$ as obtained from a complete hyperbolic orbifold by removing (isolated) cone-points.

See also my answer here.