An Undergraduate research experience for six students over seven weeks. Computational group theory methods are employed to explore the geometry of hyperbolic Riemann surfaces tiled by triangles and the induced tilings of the hyperbolic plane. All tilings considered generate large symmetry groups of the surface, and the geometrical and combinatorial properties of the tiling are strongly reflected in the structure of the group and the groupOs action on the geometry. The geometric and combinatorial problems need to be solved by massive computations in the symmetry groups, as it is very difficult to visually represent the complexities of these tilings. Motivated by these problems, students will perform group theoretic experiments, make discoveries and formulate conjectures by carrying out computer calculations using the software package MAGMA. The end goal will be to discover and prove theorems about the geometry and combinatorics of the surfaces and, of course, anything about groups discovered along the way. In addition to this technical program, students will be engaged in a companion program to develop their oral and written mathematical communication skills, collaborative, and other professional skills. This will be accomplished through required oral presentations and technical reports on their work and a close, positive working environment among the students and the P.I. in the Theorodrome, the computer laboratory/workplace devoted to the REU.
