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The geometry of compact hyperbolic surfaces with conical singularities isinvestigated. The Teichmuller space of a hyperbolic pair of pants with asingle singularity is studied. A finite number of real parameters whichdetermine the geometry of a hyperbolic torus of genus two with a singlesingularity, is given.

An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Mobius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.

For an-dimensional compact hyperbolic manifoldMn a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofMn. Its proof relies upon local density bounds for hyperbolic sphere packings.

This paper deals with filling the hyperbolic space Hn by non-compact polyhedra. In dimensions n