Simple Closed Geodesics on Γ(3) Surfaces

Question

Which closed geodesics on a Γ(3) surface have no self-intersections?

Goals

• Construct the Γ(3) surface and its fundamental domain.
• Classify or generate simple closed geodesics.

Method

Combine the group description of the surface with a geometric test for crossing.

• Generate elements or words in the congruence subgroup Γ(3).
• Draw the corresponding geodesics across identified copies of the domain.
• Detect self-intersections and organize the simple examples by length or word.