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McShane-type Identities for Affine Deformations

We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finitesimally lengthen the boundary.


Introduction
Beginning with McShane [26], sums of geometric quantities over the simple closed curves on a surface Σ have been shown to exhibit remarkable properties, such as being constant over the deformation space of geometric structures on Σ. Let F(Σ) denote the Fricke space, consisting of isotopy classes of marked complete hyperbolic structures on Σ. If γ is a closed curve on a marked hyperbolic surface X, then the function assigning to X the length of the closed geodesic on X (or 0 if γ is homotopic to a cusp) is a basic quantity, depending on the homotopy class of γ. See Abikoff [2] and Wolpert [40, §3] for background on hyperbolic Riemann surfaces, where F(Σ) idenfifies with the Teichmüller space of Σ by the uniformization theorem.
Mirzakhani [28] and Tan-Wong-Zhang [36] express the length of a boundary component as an infinite series of functions of the lengths of interior curves. (Compare also Hu-Tan-Zhang [19] and Wolpert [40, pp 86-87]) This led Mirzakhani to new calculations of the Weil-Petersson volume of the moduli space of curves.
In another direction, Labourie-McShane [23] found identities involving quantities generalizing geodesic lengths for Hitchin representations of π 1 (Σ) into SL(n, R). For n = 3, these quantities are the geodesic length functions for the Hilbert metric on convex RP 2 -structures on Σ. In contrast, our identities are the first such identities for a non-reductive Lie group.
Closely related to hyperbolic structures on sufraces are flat Lorentzian structures on 3-manifolds, that is, geometric structures modeled on Minkowski space E 2,1 and groups of (Lorentzian) isometries. A Margulis spacetime M 3 is a quotient E 2,1 /Γ, where Γ ⊂ Isom(E 2,1 ) is a discrete group of Lorentzian isometries which acts properly and freely on M . We assume Γ is not solvable, as the solvable cases are easily classified (Fried-Goldman [12]). Then M 3 is an geodesically complete flat Lorentzian manifold with free fundamental group Γ. It is necessarily orientable ( [7]). The linearization Isom + (E 2,1 ) −→ SO(2, 1) maps Γ isomorphically onto a discrete subgroup of SO(2, 1) ∼ = Isom(H 2 ) ( [12]). Therefore, associated to M 3 is a complete hyperbolic surface X = H 2 /Γ 0 . Just as essential closed curves in X are canonically homotopic to unique closed geodesics (having length X (γ)), nonparabolic closed curves in M are canonically homotopic to closed geodesics which are necessarily spacelike. These geodesics have a natural Lorentzian length α(γ), called the Margulis invariant, as this quantity was discovered and developed by Margulis [24,25]. Such a geodesic is the quotient of a line Axis(γ) upon which γ acts by translation of Lorentzian displacement α(γ). This is similar to how (γ) measures the minimal displacement of the linear part of γ acting isometrically on H 2 .
The Lie group of interest here is the tangent bundle Lie group T G (see below) of the isometry group G = Isom(H 2 ) of the hyperbolic plane H 2 . The Lie algebra g of G naturally identifies with the Lorentzian vector space R 2,1 , and Ad identifies with the isometric action of G ∼ = SO(2, 1) on R 2,1 . The affine space E 2,1 identifies with the quotient T G/0 G by the zero-section.
Given any Lie group G with Lie algebra g, the tangent bundle of G enjoys a Lie group structure as the semidirect product is the adjoint representation. Then g is an abelian normal subgroup, and the zero-section is the Zariski tangent bundle of the representation variety Hom(Γ, G). In particular homomorphisms Γ −→ T G correspond to a homomorphism Γ ρ − − → G and an infinitesimal deformation of ρ, that is, a tangent vector in T ρ Hom(Γ, G) .
We obtain an identity for affine deformations Γ of a Fuchsian group Γ 0 by differentiating the basic length identity for Γ 0 . To simplify notation, define two functions H(u, v) and K(u, v): Theorem 0.1. -Let X = H 2 /Γ 0 be a complete one-ended orientable hyperbolic surface. For any affine deformation of Γ 0 , the Margulis invariants TOME 67 (2017), FASCICULE 5 satisfy: Here C denotes the set of unordered pairs of isotopy classes of simple closed curves such that γ 1 , γ 2 and ∂X cobound an embedded 3-holed sphere in X.
Theorem 0.1 implies the following result in hyperbolic geometry: Corollary 0.2. -Suppose that X is a complete one-ended orientable hyperbolic surface. Let X t be a deformation of hyperbolic surfaces such that the derivative of the length satisfies: Yair Minsky pointed out that this corollary can also be deduced from Bestvina-Bromberg-Fujiwara-Souto [3], using the convexity of length functions with respect to shear deformations. François Guéritaud observed that this corollary also follows from the theory of strip deformations developed in his collaboration [9,10,18] with Jeffrey Danciger and Fanny Kassel. In particular, he described deformations where all the interior simple loops lengthen, and all but one boundary component lengthens.
A direct application of the technique of Labourie-McShane [23] fails, since the cross-ratios of [23] will not apply directly to the non-reductive group Isom + (E 2,1 ). However, a generalization of these cross-ratios which includes infinitesimal information may give an alternate proof of our results.

Deformations and the Margulis invariant
where E 2,1 is 3-dimensional Minkowski space, that is, a simply connected geodesically complete flat Lorentzian 3-manifold, and L is projection onto the linear part. Affine deformations of Fuchsian groups arise as holonomy groups of Margulis spacetimes, complete flat Lorentz 3-manifolds with free fundamental group.
The tangent space to the Fricke space at X identifies with H 1 (Γ, g), where Γ is the image of the holonomy representation for X and g is the Lie algebra of the group of isometries of the hyperbolic plane. Since g ∼ = R 2,1 , a tangent vector V corresponds to an affine deformation of Γ, that is, a group of affine isometries of the Minkowski spacetime R 2,1 . Given γ ∈ Γ, the affine deformation of γ acts by translation on a unique invariant spacelike line Axis(γ). The signed Lorentzian distance of this translation is called the Margulis invariant α(γ) of the affine deformation.
A deformation of hyperbolic surfaces X t is a one-parameter family of marked hyperbolic surfaces of fixed topology, varying smoothly in t ∈ [a, b], where [a, b] ∈ R is a closed interval. The space of equivalence classes of marked hyperbolic surfaces forms a smooth manifold, corresponding to a smooth submanifold of the space of representations π 1 (X) −→ SO(2, 1). Thus a deformation X t corresponds to a smooth path of holonomy representations ρ t in the top-dimensional stratum of Hom π 1 (X), SO(2, 1) . Infinitesimal deformations of X correspond to tangent vectors in this space, and also to affine deformations defined above. The Margulis invariant equals the derivative of the geodesic length function Xt (γ) at t = 0 (Goldman-Margulis [16]).

Length identities for simple geodesics
Let X be a complete hyperbolic surface with one end, which we denote ∂. The length of a closed geodesic γ in X will be written X (γ), or simply (γ), when the context is clear. When γ is peripheral, then defining X (γ) to be zero makes the function X −→ X (γ) continuous on the deformation space.
The end of X lies in various three-holed spheres such that the other two ends correspond to simple closed geodesics γ 1 , γ 2 ⊂ X. (These curves coincide when g = 1.) The set C of isotopy classes of such subsurfaces identifies with the set of subsets {γ 1 , γ 2 } of isotopy classes of simple closed curves such that ∂, γ j , γ k cobound a three-holed sphere in X. Set: where D is the continuous function: (2.2) D(x, y, z) := 2 log e x/2 + e (y+z)/2 e −x/2 + e (y+z)/2 The generalized McShane identity for one-ended orientable X is: We briefly discuss the convergence of this series, as the ideas will be crucially used when we differentiate it. The key idea is that while the number of simple closed geodesics grows polynomially, the summands in these series decay exponentially.
Break C into a countable disjoint union of finite sets C N as follows. For every nonnegative integer N , is finite with cardinality:  [34] and Wolpert [40] for further discussion. The constant m = m(g, Q) depends only on the genus g and a quantity Q defined as follows. Choose > 0 sufficiently small so that closed geodesics shorter than do not intersect, such as: compare Buser [4, §4]. The Margulis lemma and the collar lemma imply only finitely many such geodesics exist. Explicitly, any set of disjoint closed TOME 67 (2017), FASCICULE 5 geodesics extends to a pants decomposition. Let b denote the number of ends of X. Since the number of pants equals −χ where χ = 2 − 2g − b is the Euler characteristic, at most 3g − 3 + b closed geodesics are shorter than . Let Q be the product of the inverses of the lengths of these geodesics. By taking t in an interval [a, b] containing 0, the continuity of Q implies we can choose m so that inequality (2.3) holds for all t ∈ [a, b] although the finite subsets C N may vary with t.
To prove that the series (2.1) converges absolutely, rearrange its terms as follows: where f N (t) denotes the finite sum of positive terms: Lemma 2.1.

Differentiating length identities
Now differentiate the generalized McShane identity to establish uniform absolute convergence of f N (t). The partial derivatives: decay exponentially as well: Proof. -Apply (3.1) and the chain rule.
By Lemma 3.3, the coefficient on the left-hand side is negative. By (3.1), each coefficient on the right-hand side is negative also. If all α(γ) > 0 for interior curves, then α(∂) > 0 as desired. BIBLIOGRAPHY