FENCHEL NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 41, 2016, FENCHEL NIELSEN COORDINATES FOR ASYMPTOTICALLY CONFORMAL DEFORMATIONS Dragomir Šarić Queens College of CUNY, Department of Mathematics Kissena Blvd., Flushing, NY 11367, U.S.A.; The CUNY Graduate Center, Mathematics PhD Program 365 Fifth Avenue, New York, NY , U.S.A. Abstract. Let X be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessrini, Liu, Papadopoulos, Su Sun [2, 3] parametrized the quasiconformal Teichmüller space T qc (X) the length spectrum Teichmüller space T ls (X) using the Fenchel Nielsen coordinates. A quasiconformal map f: X Y is said to be asymptotically conformal if its Beltrami coefficient µ = f/ f converges to zero at infinity. The space of all asymptotically conformal maps up to homotopy post-composition by conformal maps is called little Teichmüller space T 0 (X). We find a parametrization of T 0 (X) using the Fenchel Nielsen coordinates a parametrization of the closure T 0 (X) of T 0 (X) in the length spectrum metric. We also prove that the quotients AT(X) = T qc (X)/T 0 (X), T ls (X)/T qc (X) T ls (X)/T 0 (X) are contractible in the Teichmüller metric the length spectrum metric, respectively. Finally, we show that the Wolpert s lemma on the lengths of simple closed geodesics under quasiconformal maps is not sharp. 1. Introduction Let X be a complete hyperbolic surface with infinitely generated fundamental group whose boundary, if non-empty, consists of simple closed geodesics. Let P = {α n } be a geodesic pants decomposition of X by simple closed geodesics α n, where some α n can have zero length i.e., α n can be a puncture. In addition, assume that there exists a fixed constant P > 0 such that l αn (X) P for all n, where l αn (X) is the length of the geodesic α n for the hyperbolic metric of X. We say that the pants decomposition P = {α n } is upper bounded. The quasiconformal Teichmüller spacet qc (X) consists of all quasiconformal maps f: X Y up to post-composition by isometries up to homotopies. The length spectrum Teichmüller space T ls (X) consists of all homeomorphisms h: X Y up to post-composition by isometries up to homotopies such that L(X,Y) := supmax β { lβ (X) l h(β) (Y), l h(β)(y) l β (X) } <, where the supremum is over all simple closed curvesβ onx, wherel β (X),l h(β) (Y) are the lengths of the geodesic representatives of β, h(β) on X, Y, respectively. doi: /aasfm Mathematics Subject Classification: Primary 30F60, 32G15. Key words: Teichmüller, asymptotically conformal, infinite surfaces, geodesic lenghts. This research was partially supported by National Science Foundation grant DMS

2 168 Dragomir Šarić Wolpert (cf. [11]) proved that if f: X Y is a K-quasiconformal map between hyperbolic surfaces then for each non-trivial simple closed curve β on X we have (1) 1 K l β(x) l f(β) (Y) Kl β (X) which gives T qc (X) T ls (X). Let {(l n (Y),t n (Y))} n denote the Fenchel Nielsen coordinates for a marked hyperbolic surface Y corresponding to the pants decomposition P = {α n }, where (l n (Y),t n (Y)) is defined for all n with l αn (X) > 0 α n not a boundary component. If α n is a boundary geodesic, then the twist t n (Y) is not defined. Alessrini, Liu, Papadopoulos, Su Sun [3] proved that a sequence {(l n,t n )} n with l n > 0 t n R defined for all n with l αn (X) > 0 are the Fenchel Nielsen coordinates of Y T qc (X) if only if there exists M > 0 such that logl n /l n (X) M t n t n (X) M. Moreover, the Fenchel Nielsen map given by Y {(logl n (Y),t n (Y))} n is a locally bi-lipschitz parametrization of T qc (X) equipped with the Teichmüller metric by the space of bounded sequences l equipped with the supremum norm (cf. [3]). A quasiconformal map f: X Y is said to be asymptotically conformal if for every ǫ > 0 there exists a compact set K X such that µ X K < ǫ, where µ = f/ f is the Beltrami coefficient of f. The little Teichmüller space T 0 (X) consists of all asymptotically conformal quasiconformal maps f: X Y up to homotopy post-composition by isometries. Note that T 0 (X) is a closed nowhere dense subset of T qc (X). We prove Theorem 1. Let X be an infinite hyperbolic surface equipped with an upper bounded geodesic pants decomposition P = {α n } n. Then a sequence {(l n,t n )} n with l n > 0 t n R represents a point in T 0 (X) if only if (2) log l n l n (X) 0 (3) t n t n (X) 0 as n. Remark 1. The equation (2) is the asymptotically conformal version of the Wolpert s inequality (1). by The length spectrum distance d ls between two marked surfaces X Y is defined d ls (X,Y) = 1 2 logl(x,y). Shiga [10] started the study of the length spectrum distance d ls on T qc (X) for infinite surfaces he established that d ls is not complete on T qc (X) in general. Alessrini, Liu, Papadopoulos Su [1] proved that T ls (X) is complete in the length spectrum distance d ls. Moreover, the closure of T qc (X) for the length spectrum distance is nowhere dense in T ls (X) (cf. [2]). Alessrini, Liu, Papadopoulos Su

3 Fenchel Nielsen coordinates for asymptotically conformal deformations 169 [2] proved that a sequence {(l n,t n )} n with l n > 0 t n R represents a point in T ls (X) if only if there exists M > 0 such that (4) log l n l n (X) M (5) t n t n (X) max{1, logl n (X) } M for all n. The normalized Fenchel Nielsen map defined by {( F(Y) = log l )} n(y) l n (X), t n (Y) t n (X) max{1, logl n (X) } is a locally bi-lipschitz homeomorphism from T ls (X) equipped with the lengthspectrum metric onto l equipped with the supremum norm (cf. [8]). In addition, a marked surface Y is in the closure T qc (X) of T qc (X) in the length spectrum metric if only if there exists M > 0 such that the Fenchel Nielsen coordinates {(l n (Y),t n (Y))} of Y satisfy (cf. [8]) log l n(y) l n (X) M t n (Y) t n (X) = o( logl n (X) ) as l n (X) 0. (If a sequence a n then b n = o(a n ) satisfies b n /a n 0.) We prove Theorem 2. Let X be an infinite complete hyperbolic surface with geodesic boundary, if any, that has an upper bounded geodesic pants decomposition. Let T 0 (X) be the closure in the length spectrum metric of the little Teichmüller space T 0 (X). Then Y T 0 (X) if only if its Fenchel Nielsen coordinates satisfy (6) log l n(y) l n (X) 0 (7) as n. We obtain the following corollary t n (Y) t n (X) max{1, logl n (X) } 0 Corollary 1. Let X be an infinite complete hyperbolic surface with geodesic boundary, if any, that has an upper bounded geodesic pants decomposition. Then the asymptotic Teichmüller space AT(X) = T qc (X)/T 0 (X) is contractible in the Teichmüller metric. Moreover, the quotient T ls (X)/T qc (X) T ls (X)/T 0 (X) are contractible in the length spectrum metric. Fletcher [6, 7] proved that AT(X) is locally bi-lipschitz to the space l /c 0 for the unreduced Teichmüller spaces, where l is the space of bounded sequences c 0 is the subspace of sequences converging to zero. The above parameterizations [8] give that AT(X), T ls (X)/T qc (X) T ls (X)/T 0 (X) are globally parametrized n

4 170 Dragomir Šarić locally bi-lipschitz to l /c 0 where the spaces are defined in the reduced sense, i.e. homotopies are allowed to move points on the boundary geodesics. For each α n in the pants decomposition P of X with l αn (X) > 0, let P 1 n P2 n be the two pairs of pants of P that have α n on their boundary with possibly P 1 n = P2 n. Define γ n to be a shortest closed geodesic in P 1 n P2 n intersecting α n in either one or two points. We have Proposition 1. Let X be an infinite complete hyperbolic surface whose boundary, if any, consists of simple closed geodesics. Let f: X Y be a quasiconformal map. Then sup l γn (Y) l γn (X) <. n Remark 2. The above proposition proves that the Wolpert s inequality (1) is not sharp if there exists a subsequence γ nk with l γnk (X) as n k. This is the case when l αnk (X) 0 as n k. 2. The Fenchel Nielsen coordinates for T 0 (X) Let X be an infinite surface (i.e., the fundamental group of X is infinitely generated) that is equipped with a complete hyperbolic metric such that the boundary of X, if non-empty, consists of simple closed geodesics. Assume X has a generalized geodesic pants decomposition P = {α n } n (generalized pants decomposition means that some of the boundary components α n of the pants could be punctures) such that there exists a constant P > 0 with l αn (X) P. Here l αn (X) sts for the length of the geodesic α n in the hyperbolic metric of X. If α n is a puncture, then l αn (X) = 0. We say that the geodesic pants decomposition P = {α n } is upper bounded. A quasiconformal map f: X Y is asymptotically conformal if for every ǫ > 0 there exists a compact set K ǫ X such that the Beltrami coefficient µ = f/ f satisfies µ X Kǫ < ǫ. Note that the compact set K ǫ might contain finitely many boundary geodesics of X. The quasiconformal Teichmüller space T qc (X) consists of all quasiconformal maps f: X Y up to post-composition by hyperbolic isometries of the image surface up to homotopy. Note that the homotopies are fixing boundary geodesics set-wise but they can move points on the boundary. The quasiconformal Teichmüller space T qc (X) is also known as the reduced Teichmüller space. The little Teichmüller space T 0 (X) consists of all asymptotically conformal quasiconformal maps f: X Y up to post-composition by hyperbolic isometries of the image surface up to homotopy. The homotopies set-wise fix boundary geodesics of X. Given a homeomorphism h: X Y, a closed geodesic β on X is mapped onto a closed curveh(β) ony. We denote byl h(β) (Y)the length of the unique closed geodesic iny homotopic toh(β). Ifα n is a geodesic of the pants decomposition P ofx, denote l n (Y) = l h(αn)(y) the length part of the Fenchel Nielsen coordinates for Y. If α n is a boundary geodesic then the twist t n is not defined. If α n is an interior geodesic of X, then we define the twist parameter t n (Y) as the twist parameter of a compact subsurface of Y with geodesic boundary which contains h(α n ) in its interior. It is irrelevant which subsurface we choose (cf. [2]). Therefore we obtained the Fenchel Nielsen coordinates {(l n (Y),t n (Y)} for any marked surface Y, where (l n,t n ) is not specified when l αn (X) = 0 t n is not specified when α n is a boundary geodesic.

5 Fenchel Nielsen coordinates for asymptotically conformal deformations 171 Proof of Theorem 1. Assume first that {(l n,t n )} is given that satisfies (2) (3). Denote by Y a hyperbolic surface homeomorphic to X with the Fenchel Nielsen coordinates {(l n,t n )} (cf. [2]). Note that Y is constructed by gluing geodesic pairs of pants according to the twists t n. The shape of the geodesic pants is determined by the lengths l n. Following [2], we construct a quasiconformal map f: X Y such that the surface Y has Fenchel Nielsen coordinates {l n, l n l n(x) t n(x)}. It is immediate from [2] Bishop [5] that the quasiconformal map f: X Y is asymptotically conformal by the condition (2). It remains to prove that there is an asymptotically conformal map g: Y Y. Then g f: X Y is an asymptotically conformal map. Note that each geodesic boundary α n in the interior of Y has a collar neighborhood of width bounded below by a positive constant. Thus the twisting by the amount t n ln t l n(x) n(x) along the family α n is homotopic to a quasiconformal map obtained by an explicit construction in each collar neighborhood (cf. Wolpert [11]) because t n ln t l n(x) n(x) is bounded. Moreover, conditions (2) (3) imply that t n l n t l n(x) n(x) 0 as n the construction in [11] implies that the map g can be chosen to be asymptotically conformal. It remains to prove that if f: X Y is asymptotically conformal K-quasiconformal map, then its Fenchel Nielsen coordinates {(l n (Y),t n (Y))} n satisfy (2) (3). Let H be the upper half-plane model of the hyperbolic plane. We consider a sequence of universal coveringsπn 1: H X π2 n : H Y such that positive half of they axis covers the geodesic α n X the geodesic homotopic to f(α n ) Y, respectively. Let f n : H H be the lift of f: X Y that fixes 0, 1. Denote by A 1 n A 2 n the primitive hyperbolic elements of the covering groups for π1 n π2 n for the curves α n X f(α n ) Y, respectively. Then f n A 1 n f 1 n = A 2 n the translation lengths of A 1 n A2 n are l n(x) l n (Y), respectively. Then K P is an upper bound for the lengths of the pants decomposition {f(α n )} on Y. Let a n N be such that P a n l n (X) 2P. To prove (2) assume that it is false find a contradiction. Namely there exists a subsequence l nk (Y) C > 0 such that (8) log l n k (Y) l nk (X) C > 0. The sequence of quasiconformal maps f nk : H H has the same quasiconformal constant it fixes 0, 1. Therefore it has a subsequence that converges uniformly on compact subsets of H to a quasiconformal map f: H H for simplicity of notation denote this subsequence by f nk. The sequence of Beltrami coefficients µ nk = f nk / f nk converges to zero pointwise a.e. on H as n k because f nk is a lift of an asymptotically conformal quasiconformal map f such that α nk lifts to the positive y axis. Therefore f is a conformal map which fixes 0, 1, therefore f = id. Recall thata 1 n k A 2 n k are primitive elements of the covering Fuchsian groups of X Y that are lifts ofα nk f nk (α nk ). Recall further that a nk N is chosen such that P a nk l nk (X) 2P. Denote by (A 1 n k ) an k (A 2 n k ) an k the a nk -compositions of A 1 n A2 n with themselves note that they both fix 0. They have translation lengths between 1 P 2KP. Therefore there exist subsequences of K (A 1 n k ) an k (A 2 n k ) an k which pointwise converge to hyperbolic isometries A 1 A 2

6 172 Dragomir Šarić both different from the identity. Since f nk id as n k, we have f nk (A 1 n k ) an k f 1 n k = (A 2 n k ) an k A 1 = A 2. In particular they have the same translation lengths. This implies that the ratio of the translation length of (A 1 n k ) an k to the translation length of (A 2 n k ) an k converges to 1, namely a nk l nk (Y) a nk l nk (X) = l n k (Y) l nk (X) 1 as n k. This contradicts (8) establishes (2). Let Y be the marked hyperbolic surface whose Fenchel Nielsen coordinates are l {l n, n t l n(x) n(x)}. There exists a marking map g: X Y that is an asymptotically conformal quasiconformal map by (2) (3), by the first part of this proof. Then the map f 1 = f g 1 : Y Y is an asymptotically conformal K 1 -quasiconformal map. To prove (3), assume on the contrary that there exist a subsequence t nk (Y) C > 0 such that (9) t n k (Y) l n l n (X) t n(x) = t n k (Y) t nk (Y ) C > 0 for all n k. We seek a contradiction with (9). For α n g: X Y, the geodesic on Y homotopic tog(α n ) is denoted byg(α n ) for simplicity. Denote byp the corresponding geodesic pants decomposition of Y. Let γ nk be a shortest simple closed geodesic in Y which intersects g(α nk ) is contained in the union of two geodesic pairs of pants of P with the geodesic g(α nk ) on their boundaries. We fix universal coverings π 1 n k : H Y π 2 n k : H Y such that lifts of the geodesic g(α nk ) of Y geodesic of Y homotopic to f(α nk ) contain the positive y axis, that a lift γ nk of γ nk has one endpoint at 1 R. The map f 1 : Y Y represents twisting along the family P. Let P be the lift to H of the pants decomposition P of Y. We choose a lift f nk : H H of f 1 that fixes 0, that is further normalized such that f nk fixes the ideal points on R = H of the component of H P on the left side of the positive y axis adjacent to the positive y axis. We define the cross-ratio of four points a,b,c,d R by cr(a,b,c,d) = (a c)(b d) (a d)(b c). Note that γ nk intersect only the lifts of g(α nk ) from all the geodesics in the pants decomposition P of Y. Therefore the twisting along γ nk is in the same direction (cf. Alessrini, Liu, Papadopoulos Su [2]). Let x nk < 0 be the other endpoint of γ nk. Then there exists J > 1 such that 1 J < logcr(x n k,0,1, ) = log 1 x n k x nk ( ) 1 = log +1 < J x nk for all n k by the choice of α nk γ nk, by the upper bound on the lengths of α n (cf. [3]). In particular, x nk is contained in a compact subset of R. Assume without loss of generality that t n (Y) ln l n(x) t n(x) > 0. By the choice of the lift f nk,

7 Fenchel Nielsen coordinates for asymptotically conformal deformations 173 we have that f nk (0) = 0, fnk ( ) =, fnk (1) e tn ln (Y) k ln(x) tn k (X) e C > 1 f nk (x nk ) x nk. It follows that ( ) logcr( f nk (x nk ), f nk (0), f nk (1), f e C nk ( )) log +1. x nk Thus there exists C > 0 such that (10) log cr( f nk (x nk ), f nk (0), f nk (1), f nk ( )) cr(x nk,0,1, ) C > 0 for all n k. On the other h, fnk fixes 0, f nk (1) is bounded away from 0 since the total twisting along γ nk is Thurston bounded (cf. [9]). We reflect f nk in R to obtain a quasiconformal mapping of C. Since the Beltrami coefficient of fnk converges pointwise a.e. to zero, it follows that f nk converges to the identity uniformly on compact subsets of C. This contradicts (10) proves (3). 3. The closure of T 0 (X) We consider the closure T 0 (X) of T 0 (X) as a subset of T ls (X) equipped with the length spectrum metric find its characterization in terms of Fenchel Nielsen coordinates. Proof of Theorem 2. Consider a sequence of marked surfaces Y k T 0 (X) that converges to Y T 0 (X). Let {(l n (Y k ),t n (Y k ))} be the Fenchel Nielsen coordinates of Y k. By Theorem 1, we have log l n(y k ) l n (X) 0 t n (Y k ) t n (X) 0 as n for each fixed k. Since the normalized Fenchel Nielsen map is homeomorphism for the length spectrum distance, the above conditions imply that the limit Y satisfies (6) (7). Assume now that Y T qc (X) has Fenchel Nielsen coordinates satisfying (6) (7). We need to find a sequence Y k T 0 (X) that converges to Y as k. Let Y k be defined by the Fenchel Nielsen coordinates satisfying for n k, l n (Y k ) = l n (Y), t n (Y k ) = t n (Y) l n (Y k ) = l n (X), t n (Y k ) = t n (X) for n > k. Then Y k T 0 (X) by Theorem 1. Since l n (Y k ) = l n (Y) for n k l lim n(y) n l n(x) = 1, we have that sup n log ln(y k) 0 as k. Since t l n(y) n(y k ) = t n (X) for n k, since t n (Y k ) t n (Y) = t n (X) t n (Y) for n > k, since t n (X) t n (Y) = o( logl n (X) ) (by t n (Y) T qc (X)), we get that Y k Y as k in the length spectrum metric. Proof of Corollary 1. The Fenchel Nielsen coordinates map T qc (X) homeomorphically onto l they map T 0 (X) homeomorphically onto the space c 0 of sequences that vanish at infinity. Then AT(X) is homeomorphic to l /c 0 thus contractible.

8 174 Dragomir Šarić The normalized Fenchel Nielsen coordinates map T ls (X) homeomorphically onto l they map T 0 (X) homeomorphically onto the space c 0 of sequences that vanish at infinity. Then T ls (X)/T 0 (X) is homeomorphic to l /c 0 thus contractible. To obtain the contractibility for T ls (X)/T qc (X), we separate the length the twist parts of the Fenchel Nielsen coordinates. Note that the length parts for both T ls (X) T qc (X) cover l that the quotient of normalized twist parts is l /c 0 thus contractible. 4. The lengths of closed geodesics under quasiconformal maps Wolpert [11] proved that a K-quasiconformal map distorts the lengths of simple closed geodesics by at most multiplying them with K or at least multiplying them with 1/K. We prove that this estimate is not sharp for infinite surfaces. To do so, we find a sequence of simple closed curves with lengths going to infinity such that a K-quasiconformal map changes their lengths by at most an additive constant. Proof of Proposition 1. Let l αn (X) > 0 let γ n be a shortest simple closed geodesic intersecting α n as before. Then γ n is contained in the union of two geodesic pairs of pants Pn 1 Pn 2 of the pants decomposition P = {α n } of X. Assume first that Pn 1 P2 n. Divide each pair of pants Pi n into two congruent right angled hexagons Σ i n,1 Σi n,2 by three simple mutually disjoint geodesic arcs orthogonal to pairs of boundary geodesics of Pn i (cf. Figure 1). If a boundary component of Pn i is a puncture then arcs ending at the puncture have infinite length there is no requirement to be orthogonal at this boundary component. Figure 1. Estimating the length of γ n. Let γn,1 i be the geodesic arc perpendicular to the side of Σ i n,1 contained in α n to the opposite side of the hexagon Σ i n,1. Define γn,2 i to be the geodesic arc perpendicular to the side of Σ i n,2 contained in α n the opposite boundary side of Σ i n,2. Then γn i = γn,1 γ i n,2 i is a geodesic arc in Pn i with both endpoints at α n that is also orthogonal to α n at these points (cf. Figure 1). The geodesic arc γn,1 i divides Σi n,1 into two right angled pentagons. The hexagon Σ i n,1 has one boundary side equal to half of the geodesic α n. It follows that at least one of the two pentagons has boundary side a adjacent to γn,1 i equal to at least 1/4 of the the length of the geodesic α n ( less than 1/2 of α n ). Note that this pentagon has 1/2 of a geodesic α n in P on its boundary that the length of this boundary

9 side is bounded by P 2 Since sinha a such that Moreover, we have Fenchel Nielsen coordinates for asymptotically conformal deformations 175 from the above. Then [4] we have cosh l α n (X) 2 = sinhl γ i n,1 (X)sinha. 1 as a 0, it follows that there exists C 1,C 2 > 0 depending on C 1 a sinha C 2 a. 1 cosh l α n (X) cosh P 2 2 = C 3. The above two inequalities imply that there exist C 4,C 5 > 0 such that which implies that for some C 6 > 0 C 4 e l γ i n,1(x) lαn (X) C 5 l γ i n,1 (X) logl αn C 6. Thus the lengthl γ i n (X) is up to a bounded additive constant equal to2 logl αn (X) for i = 1,2. The geodesic γ n is homotopic to the union of γ 1 n γ 2 n two sub-arcs of α n. It follows that there exist two constants C 7,C 8 R such that, for all n, C 7 +4 logl αn (X) l γn (X) C 8 +4 logl αn (X). A K-quasiconformal map f: X Y changes the lengths of l αn (X) by at most a multiplicative constant between 1/K K. However, this only changes the additive constants in the above inequality. Since l αnk (X) 0 as n k 0, we get that l γn (Y) l γn (X) C for some C > 0. Assume next that P 1 n = P2 n = P n. Let γ n be the simple geodesic arc in P n which is orthogonal to α n at both endpoints. Then P n is divided into two hexagons by drawing two more simple mutually non-intersecting geodesic arcs orthogonal to α n the other boundary component α n of P n. Fix one hexagon Σ n denote by α n,1 α n,2 the two boundary sides of Σ n that lie on α n ( therefore are adjacent to γ n). Denote by α n,3 the boundary side of Σ n on α n ( therefore opposite to γ n). Then by [4] we have coshl γ n (X)sinhl αn,1 (X)sinhl αn,2 (X) = coshl αn,3 (X)+coshl αn,1 (X)coshl αn,2 (X). Since l αn,1 (X) = l αn,2 (X) = lαn (X) 2 P/2 l αn,3 (X) P, the above equality implies that there exist C 1,C 2 > 0 such that C 1 e l γ n (X) l αn (X) 2 C 2 which implies the result as in the previous case. References [1] Alessrini, D., L. Liu, A. Papadopoulos, W. Su: On various Teichmüller spaces of a surface of infinite topological type. - Proc. Amer. Math. Soc. 140, 2012, [2] Alessrini, D., L. Liu, A. Papadopoulos, W. Su: On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space. - Preprint, available on arxiv. [3] Alessrini, D., L. Liu, A. Papadopoulos, W. Su, Z. Sun: On Fenchel Nielsen coordinates on Teichmüller spaces of surfaces of infinite-type. - Ann. Acad. Sci. Fenn. Math. 36:2, 2011,

10 176 Dragomir Šarić [4] Beardon, A.: The geometry of discrete groups. - Grad. Texts in Math. 91, Springer-Verlag, New York, [5] Bishop, C.: Quasiconformal mappings of Y-pieces. - Rev. Mat. Iberoamericana 18:3, 2002, [6] Fletcher, A.: Local rigidity of infinite-dimensional Teichmüller spaces. - J. London Math. Soc. (2) 74:1, 2006, [7] Fletcher, A.: On Asymptotic Teichmüller space. - Trans. Amer. Math. Soc. 362, 2010, [8] Šarić, D.: Fenchel Nielsen coordinates on upper bounded pants decompositions. - Math. Proc. Camb. Phil. Soc. (to appear). [9] Šarić, D.: Real complex earthquakes. - Trans. Amer. Math. Soc. 358:1, 2006, [10] Shiga, H.: On a distance defined by the length spectrum of Teichmüller space. - Ann. Acad. Sci. Fenn. Math. 28:2, 2003, [11] Wolpert, S.: The Fenchel Nielsen deformation. - Ann. of Math. (2) 115:3, 1982, Received 22 February 2015 Accepted 20 July 2015