It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.

A family of coordinates h for the Teichmller space of a compact surface with boundary was introduced by the second author. Mondello showed that the coordinate 0 can be used to produce a natural cell decomposition of the Teichmller space, invariant under the action of the mapping class group. In this paper, we show that, for any h 0, the coordinate h produces a natural cell decomposition of the Teichmller space.

For fixed subgroups Fix(ϕ) of automorphisms ϕ on hyperbolic 3-manifold groups π1(M), we observed that rk(Fix(ϕ))<2rk(π1(M)) and the constant 2 in the inequality is sharp; we also classify all possible groups Fix(ϕ).

A spherical polyhedral surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedral surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedral surfaces in terms of the triangulation and the angle assignment is established.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.