Closed Geodesics on Modular Surfaces

A guided research project in Geometry & Dynamics.

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Experience Research Geometry & Dynamics

Question

Which geodesic paths return to where they started on a modular surface?

A path that leaves one side of the domain can re-enter through an identified side and eventually close.
A path that leaves one side of the domain can re-enter through an identified side and eventually close.
A hyperbolic triangle and geodesic arcs in a Poincaré disk.
A Poincaré-disk construction from the hyperbolic-geometry research materials.

Goals

  • Connect closed geodesics with matrices in the modular group.
  • Find and draw examples inside a fundamental region.

Method

Follow a hyperbolic geodesic while identifying points related by modular transformations.

  • Work in the upper-half-plane model and construct a fundamental domain.
  • Represent side crossings by Möbius transformations or symbolic words.
  • Search for returning paths and verify closure from the associated matrix.