Immersed surfaces in the modular orbifold

Abstract

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.

Stable immersions in orbifolds

“…Theorem 4.1 generalizes [5], Theorem 3.1. The situation of interest in [5] is (2, p, ∞) orbifolds, which have only two orbifold points, and as stated, Theorem 4.1 requires 3 orbifold points. However, in the special case of two orbifold points with one point of order 2, Lemma 4.6 can be avoided, and Theorem 4.1 still goes through.…”

“…Results. In this paper, we generalize [5], Theorem 3.1 (see Remark 4.7, which addresses the issue of two vs three orbifold points) with the following theorem.…”

“…picture. For a more thorough scl background, especially as it relates to quasimorphisms and immersions, see [1] and [5].…”

Combinatorial Growth in the Modular Group

“…The study of asymptotic growth rates of geometric lengths of various classes of closed geodesics has a long and storied history beginning with Huber's result for all closed geodesics, to Mirzakhani's growth rate of the simple closed geodesics, to more general results for non-simple closed geodesics and reciprocal geodesics [1,2,3,4,8,9,13,17]. Concurrently there is the study of such geodesics in terms of word length or equivalently primitive conjugacy classes and their word length growth rates leading to more abstract, algebraic investigations of groups such as surface groups or free groups [5,7,10,15,16,18].…”

Modular cocycles and linking numbers

“…The image of this orbit in M is a modular knot. Ghys [20] gave the beautiful result that the linking number [18] of this knot with the trefoil (with some orientation) is given by the Rademacher symbol The Rademacher symbol defined for all γ ∈ Γ by (1.1) is a conjugacy class invariant [35] and, for γ hyperbolic, it is the homogenization of the Dedekind symbol Φ(γ) [5] [9]. More precisely, (1.4) Ψ(γ) = lim n→∞ Φ(γ n ) n In addition to its role here, the Dedekind sum s(a, c) occurs in surprisingly diverse contexts (see e.g.…”