**MTH ****520: ****Introduction to Hyperbolic Geometry ****(4)**

**Pre-requisites (Desirable)**: MTH 304, MTH 407

*Learning Objectives*:

Hyperbolic geometry is arguably the most important area in modern geometry and topology. This course is intended to expose the student to the foundational concepts in hyperbolic geometry, and is specially tailored to prepare the student for advance topics in geometric topology.

*Course Contents*:

**The general Möbius group.** The extended complex plane (or the Riemman sphere) **C**; The general Möbius group **Mob(Ĉ)**; Identifying **Mob ^{}^{+}(Ĉ)** with the matrix group

**PGL(2; C);**Classification of elements of elements of

**Mob**; Reflections and the general Möbius group

^{+}(Ĉ)**Mob(Ĉ)**; Conformality of elements in

**Mob(Ĉ)**.

**The upper-half plane model H ^{2}. **The upper half plane

**H**; The subgroup

^{2}**Mod(H2)**; Transitivity properties of

**Mob**); Geometry of the action of

^{+}(H^{2}**Mob**); The metric in

^{+}(H^{2}**H**; Element of arc-length in

^{2}**H**; Path metric in

^{2}**H**; The Poincaré metric

^{2}**d**on

_{H}**H**; Geodesics in

^{2}**H**; Identifying the group

^{2}**Mob**) of isometries of (

^{+}(H^{2}**H**

^{2},**d**with

_{H})**PSL(2; R)**; Ultraparallel lines in

**H**.

^{2} **The Poincaré disk model D. **The Poincaré disk **D**; Transitioning from**H ^{2}**to D via

**Mob**); Element of arc-length and the metric

^{+}(H^{2}**d**in

_{D}**D**; The Group

**Mob(D)**of isometries of

**(D, d**; Centre, radii, and length of hyperbolic circles in

_{D})**D**; Hyperbolic structures on holomorphic disks.

**Properties of H ^{2}.** Curvature of

**H**; Convex subsets of

^{2}**H**; Hyperbolic polygons; Area of a subset of

^{2}**H**; Gauss-Bonnet formula - area of a hyperbolic triangle; Applications of Gauss-Bonnet Formula: Area of reasonable hyperbolic polygons, existence of certain hyperbolic

^{2}*n*-gons, hyperbolic dilations; Putting a hyperbolic structure on a surface using hyperbolic polygons; Hyperbolic trigonometry: triogometric identities, law of sines and cosines, Pythagorean theorem.

**Non-planar models (if time permits).**Hyperboloid model for the hyperbolic plane; Higher dimensional hyperbolic spaces.

*Suggested Books*:

- James W. Anderson,
*Hyperbolic Geometry*(2nd Edition), Springer, 2005. - Arlan Ramsay, Robert D. Richtmyer,
*Introduction to Hyperbolic Geometry*, Springer, 1995. - Harold E. Wolfe,
*Introduction to Non-Euclidean Geometry*, Dover, 2012 - Alan F. Beardon,
*The geometry of discrete groups*(Chapter 7), Springer, 1983. - Svetlana Katok,
*Fuchsian Groups*(Chapter 1), Chicago Lectures in Mathematics, 1992. - John Stillwell,
*Geometry of surfaces*(Chapter 4), Springer, 1992.

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