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

## Tags » Hyperbolic Geometry

#### 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

#### 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

#### Ergodicity of the Geodesic Flow

### The geodesic flow

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

#### 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