Tags » Hyperbolic Geometry

Isomorphic groups via hyperbolic geometry

is the isometry group of hyperbolic 2-space under the hyperboloid model; is the isometry group of the upper half-plane; is the isometry group of the Poincaré unit disk. 58 more words


Mostow Rigidity: several proofs

Mostow rigidity (or, technically, the special case thereof, for , states that if M and N are two closed (i.e. compact and boundary-less) hyperbolic 3-manifolds, and is a homotopy equivalence. 1,185 more words

Hyperbolic Geometry

Arguments using moduli spaces: some examples

This blogpost started as a response to the question “why moduli spaces?” … but then David Ben-Zvi had a good pithy answer to that in his   2,278 more words

Hyperbolic Geometry

SnapPy 2.4 released

A new version of SnapPy, a program for studying the topology and geometry of 3-manifolds, is available.  Added features include a census of Platonic manifolds, rigorous computation of cusp translations, and substantial improvements to its link diagram component.


Ergodicity of the Geodesic Flow

The geodesic flow

Given any Riemannian manifold M, we may define a geodesic flow  on the unit tangent bundle which sends a point  1,058 more words

Hyperbolic Geometry

Hyperbolic Geometry Note #2: On Möbius transformations as isometries of the upper half plane

Möbius transformations are conformal (i.e., angle-preserving), orientation-preserving, area-preserving isometries in the upper half-plane model of hyperbolic space. It piqued my interest to discover that the form of the metric in this model (which I discussed in… 732 more words

Hyperbolic Geometry