16 research outputs found
Generalized Teichm\"{u}ller space of non-compact 3-manifolds and Mostow rigidity
Consider a 3dimensional manifold obtained by gluing a finite number of
ideal hyperbolic tetrahedra via isometries along their faces. By varying the
isometry type of each tetrahedron but keeping fixed the gluing pattern we
define a space of complete hyperbolic metrics on with cone
singularities along the edges of the tetrahedra. We prove that is
homeomorphic to a Euclidean space and we compute its dimension. By means of
examples, we examine if the elements of are uniquely determined
by the angles around the edges of Comment: 15 pages, 7 figure
On the Mapping class group of a genus 2 handlebody
A complex of incompressible surfaces in a handlebody is constructed so that
it contains, as a subcomplex, the complex of curves of the boundary of the
handlebody. For genus 2 handlebodies, the group of automorphisms of this
complex is used to characterize the mapping class group of the handlebody. In
particular, it is shown that all automorphisms of the complex of incompressible
surfaces are geometric, that is, induced by a homeomorphism of the handlebody
Geometries on Polygons in the unit disc
For a family of properly embedded curves in the 2-dimensional
disk satisfying certain uniqueness properties, we consider
convex polygons and define a metric on such
that is a geodesically complete metric space whose geodesics are
precisely the curves
Moreover, in the special case consists of all Euclidean lines,
it is shown that with this new metric is not isometric to any convex domain
in equipped with its Hilbert metric.
We generalize this construction to certain classes of uniquely geodesic
metric spaces homeomorphic to Comment: To appear in Rocky Mountain J. Mat