Geometric and topological invariants of curves on hyperbolic surfaces

Author

• Sayantika Mondal, CUNY Graduate CenterFollow

Date of Degree

9-2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Mathematics

Advisor

Ara Basmajian

Committee Members

Ara Basmajian

Dragomir Saric

Nick Vlamis

Subject Categories

Geometry and Topology | Mathematics

Keywords

Curve counting, Curves on surfaces, Hyperbolic geometry, Invariants, Length spectrum, Mapping class group, Teichmuller theory

Abstract

This thesis studies filling curves on closed hyperbolic surfaces along two themes. The first is enumerative: we count the mapping-class-group orbits of minimal filling curve systems. Through a correspondence with unicellular ribbon graphs we classify these orbits and, using the character theory of the symmetric group we obtain a precise asymptotic count. The second theme compares two invariants of a filling curve: its self-intersection number and its inf invariant. The self-intersection number i(γ, γ) is the standard measure of com- plexity, but it is coarse—at each value k it is shared by on the order of k^6g−7 orbits. We study a finer invariant, the inf invariant mᵧ, defined as the infimum of geodesic length over Teichmüller space, together with the optimal metric Xᵧ at which the infimum is attained. For a filling curve, the length function is proper on moduli space, so mᵧ is realized at a unique interior metric with positive systole; this fails for non-filling curves. To separate the two invariants we construct, by surgery on the minimal filling curves, families of curves with the same self-intersection number k but different inf invariants— growing at order log k in one family and at order √k in the other. Thus, mγ distinguishes orbits that the self-intersection number identifies. On cusped surfaces, the separation extends to chains of curves whose length grows at least linearly with the genus. We track the position of the optimal metric in moduli space through its systole, one family staying compact while the other degenerates to a boundary stratum, and we define the inf spectrum and provide coarse bounds on its growth.

Recommended Citation

Mondal, Sayantika, "Geometric and topological invariants of curves on hyperbolic surfaces" (2026). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/6885