MTH418: Fuchsian groups

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Several models of the hyperbolic space: the upper-half space model and the Poincaré disc model.

• Hyperbolic distance, area and geodesics.

• The Möbius group: action of ${\mathrm PSL}(2 , {\mathbb{R}})$ on the hyperbolic space.

• Classifying different types of isometries.

• Hyperbolic triangles, hyperbolic trigonometry, hyperbolic polygons.

• Fuchsian groups: discrete subgroups of ${\mathrm PSL}(2 , {\mathbb{R}})$.

• Fundamental domains and Dirichlet regions.
• Limit sets. Elementary and non-elementary Fuchsian groups.

• Poincaré's theorem and groups generated by side-pairing transformations.

Recommended Reading

• S. Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press.

• J. Anderson, Hyperbolic Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, 1999.

• Alan F. Beardon, The Geometry Of Discrete Groups, Graduate Texts in Mathematics 91, Springer-Verlag, 1983.

• John G. Ratcliffe, Foundation Of Hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer-Verlag, 1994