Immersed surfaces in the modular orbifold

TL;DR

This paper investigates the conditions under which hyperbolic conjugacy classes in the modular group correspond to geodesics that virtually bound immersed surfaces in the modular orbifold, revealing a stability property involving parabolic elements.

Contribution

It proves a stability theorem showing that multiplying hyperbolic elements by large powers of parabolic elements yields geodesics that virtually bound immersed surfaces.

Findings

Hyperbolic conjugacy classes can be associated with geodesics that virtually bound immersed surfaces.

A stability theorem is established for hyperbolic elements combined with large powers of parabolic elements.

The polyhedral structure in the stable commutator length norm relates to the immersed surface property.

Abstract

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.