Immersed surfaces in the modular orbifold

Calegari, Danny; Louwsma, Joel

doi:10.1090/S0002-9939-2011-10911-0

Immersed surfaces in the modular orbifold
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by Danny Calegari and Joel Louwsma PDF

Proc. Amer. Math. Soc. 139 (2011), 2295-2308 Request permission

Abstract:

A hyperbolic conjugacy class in the modular group $\mathrm {PSL}(2,\mathbb {Z})$ 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.

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