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Immersed Surfaces in the Modular Orbifold 1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 7, July 2011, Pages 2295–2308 S 0002-9939(2011)10911-0 Article electronically published on March 7, 2011 IMMERSED SURFACES IN THE MODULAR ORBIFOLD DANNY CALEGARI AND JOEL LOUWSMA (Communicated by Alexander N. Dranishnikov) Abstract. A hyperbolic conjugacy class in the modular group PSL(2, Z)cor- responds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is re- lated 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. 1. Introduction In many areas of geometry, it is important to understand which immersed curves on a surface bound immersed subsurfaces. Such questions arise (for example) in topology, complex analysis, contact geometry and string theory. In [5] it was shown that studying isometric immersions between hyperbolic surfaces with geo- desic boundary gives insight into the polyhedral structure of the second bounded cohomology of a free group (through its connection to the stable commutator length norm, defined in §2.2). If Σ is a (complete) noncompact oriented hyperbolic orbifold, a (hyperbolic) con- jugacy class g in π1(Σ) is represented by a unique geodesic γ on Σ. The immersion problem asks when there is an oriented surface S and an orientation-preserving im- mersion i : S → Σ taking ∂S to γ in an orientation-preserving way. If the problem has a positive solution one says that γ bounds an immersed surface. The immer- sion problem is complicated by the fact that there are examples of curves γ that do not bound immersed surfaces but have finite (possibly disconnected) covers that do bound; i.e. there is an immersion i : S → Σasaboveforwhich∂S → γ factors through a covering map of some positive degree. In this case we say γ virtually bounds an immersed surface. One can extend the virtual immersion problem in a natural way to finite rational formal sums of geodesics representing zero in (rational) homology. Formally, one H § defines the real vector space B1 (G)ofhomogenized 1-boundaries (see 2.2), where H G = π1(Σ). It is shown in [5] that the set of rational chains in B1 for which the virtual immersion problem has a positive solution are precisely the set of rational points in a closed convex rational polyhedral cone with nonempty interior. This fact is connected in a deep way with Thurston’s characterization (see [15]) of the Received by the editors April 19, 2010. 2010 Mathematics Subject Classification. Primary 20F65, 20H10, 57M07. c 2011 American Mathematical Society Reverts to public domain 28 years from publication 2295 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2296 DANNY CALEGARI AND JOEL LOUWSMA set of classes in H2(M; Z) represented by fibers of fibrations of a given 3-manifold M over S1. It can be used to give a new proof of symplectic rigidity theorems of Burger-Iozzi-Wienhard [3], and it has been used by Wilton [16] in his work on Casson’s conjecture. Evidently understanding the structure of the set of solutions of the virtual immersion problem is important, with potential applications to many areas of mathematics. Unfortunately, this set is apparently very complicated, even for very simple surfaces Σ. The purpose of this paper is to prove the following Stability Theorem: Stability Theorem 3.1. Let v be any hyperbolic conjugacy class in PSL(2, Z), represented by a string X of positive R’s and L’s. Then for all sufficiently large n, the geodesic in the modular orbifold M corresponding to the stabilization RnX virtually bounds an immersed surface in M. This theorem proves the natural analogue of Conjecture 3.16 from [5], with PSL(2, Z)inplaceofthefreegroupF2. It follows from the main theorems of [5] that the elements vn ∈ PSL(2, Z) corre- sponding to the stabilizations of v as above satisfy scl(vn)=rot(vn)/2, where rot is the rotation quasimorphism on PSL(2, Z) and scl denotes the stable commutator length (see §2 for details). Under the natural central extension φ : B3 → PSL(2, Z) where B3 denotes the 3 strand braid group, there is an equality scl(b)=scl(φ(b)) for all b ∈ [B3,B3]. Consequently we derive an analogous stability theorem for stable commutator length in B3. We give the necessary background and motivation in §2. Theorem 3.1 is proved in §3. In §4 we generalize our main theorem to (2,p,∞)-orbifolds for any p ≥ 3, and we discuss some related combinatorial problems and a connection to a problem in theoretical computer science. Finally, in §5 we describe the results of some computer experiments. 2. Background We recall here some standard definitions and facts for the convenience of the reader. 2.1. The modular group. The modular group PSL(2, Z) acts discretely and with finite covolume on H2, and the quotient is the modular orbifold M,whichcanbe thought of as a triangle orbifold of type (2, 3, ∞). Every element of PSL(2, Z) either has finite order or is parabolic or is conjugate to a b a am a product of the form R 1 L 1 R 2 ···L ,wheretheai and bi are positive integers, and where L and R are represented by the matrices 10 11 L = R = 11 01 (the parabolic elements are conjugate into the form Ra or Lb). The group PSL(2, Z) is abstractly isomorphic to the free product Z/2Z ∗ Z/3Z. 2.2. Stable commutator length. For a basic introduction to stable commutator length, see [6], especially Chapters 2 and 4. If G is a group and g ∈ [G, G], the commutator length of g (denoted cl(g)) is the smallest number of commutators in G whose product is g,andthestable n commutator length scl(g) is the limit scl(g) := limn→∞ cl(g )/n. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use IMMERSED SURFACES IN THE MODULAR ORBIFOLD 2297 Stable commutator length extends in a natural way to a function on B1(G), the space of real group 1-boundaries (i.e. real group 1-chains representing 0 in homology with real coefficients) and descends to a pseudo-norm (which can be H thought of as a kind of relative Gromov-Thurston norm) on the quotient B1 (G):= −1 n B1(G)/g − hgh ,g − ng. The dual of this space (up to scaling by a factor of 2) is Q(G)/H1(G) — i.e. homogeneous quasimorphisms on G modulo homomorphisms, with the defect norm. This duality theorem, known as Generalized Bavard Duality,isprovedin[6],§2.6. A special case of this theorem, proved by Bavard in [2], says that for any g ∈ [G, G] there is an equality scl(g)=supφ φ(g)/2 where the supremum is taken over all homogeneous quasimorphisms φ ∈ Q(G) normalized to have defect D(φ)equalto 1. A quasimorphism φ (with D(φ)=1)isextremal for g if scl(g)=φ(g)/2. The theory of stable commutator length has deep connections to dynamics, group theory, geometry, and topology. However, although in principle this function con- tains a great deal of information, it is notoriously difficult to extract this information and to interpret it geometrically. It is a fundamental question in any group G to cal- H culate scl on chains in B1 (G), and to determine extremal quasimorphisms for such elements. Conversely, given a homogeneous quasimorphism φ ∈ Q(G) (especially one with some geometric “meaning”), it is a fundamental question to determine H the(possiblyempty)coneinB1 (G)onwhichφ is extremal. 2.3. Rotation quasimorphism. If G is a word-hyperbolic group, scl is a genuine H norm on B1 (G). Moreover, it is shown in [4] and [5] that if G is virtually free, the unit ball in this norm is a rational polyhedron, and there are codimension one faces associated to realizations of G as the fundamental group of a complete oriented hyperbolic orbifold. Dual to each such codimension one face is a unique extremal vertex of the unit ball in Q/H1. In our case, PSL(2, Z) may be naturally identified with the funda- mental group of the modular orbifold. The unique homogeneous quasimorphism dual to this realization, scaled to have defect 1, is the rotation quasimorphism, denoted rot. This rotation function is very closely related to the Rademacher ϕ function, which arises in connection with Dedekind’s η function and is studied by many authors, e.g. [1, 10, 12, 14, 11] and so on (see especially [10], §3.2, for a discussion most closely connected to the point of view of this paper). In fact, up to a constant, the rotation quasimorphism is the homogenization of the Rademacher n function; i.e. rot(g) = limn→∞ ϕ(g )/6n. The simplest way to define this function (at least on hyperbolic elements of PSL(2, Z)) is as follows. Associated to a hyperbolic conjugacy class g ∈ PSL(2, Z) is a geodesic γ on M.Thegeodesicγ cuts M up into complementary regions Ri (see Figure 1 for an example). Join each region Ri to the cusp by a proper ∩ ray αi, and define ni to be the signed intersection number ni = αi γ.Then 1 rot(g)= 2π i ni area(Ri).
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