##
**Amenability, Poincaré series and quasiconformal maps.**
*(English)*
Zbl 0672.30017

Let X be a Riemann surface. X is hyperbolic if it is covered by the unit disk and X is of finite type if it is obtained from a compact surface by deleting a finite number of points. Let Q(X) be the space of holomorphic quadratic differentials \(\phi (z)dz^ 2\) on X, such that
\[
\| \phi \| =\iint_{X}| \phi (z)| dx dy<\infty.
\]
Let \(B_ X\) denote the open unit ball in the Banach space Q(X).

Now suppose f: \(Y\to X\) is a covering map. Then there is a natural operator \(\Theta\) : Q(Y)\(\to Q(X)\) defined by pulling back over the branches of \(f^{-1}\) and summing over all of the branches. This is the classical Poincaré theta operator when f is the universal covering of the unit disk onto a Riemann surface X. As an operator between the Banach spaces Q(Y) and Q(X), the norm of \(\Theta\) is less than or equal to 1. The chief result of this paper describes the types of coverings f for which the norm of \(\Theta\) is strictly less than 1. Say a covering f is amenable if there are large balls with small boundary in a graphic caricature of the covering. If X is hyperbolic and of finite type, then its universal cover is nonamenable.

Theorem. Let \(Y\to X\) be a covering of a hyperbolic Riemann surface X. Either:

1. The covering is amenable, and \(\Theta (B_ Y)=B_ Y\), or

2. The covering is nonamenable, and the closure of \(\Theta (B_ Y)\) is contained in the interior of \(B_ X.\)

As a corollary, if G is a nonabelian Fuchsian group acting on the unit disk \(\Delta\) such that \(X=\Delta /G\) is a compact Riemann surface of finite type, then the norm of \(\Theta\) is less than 1 and the inclusion mapping from the Teichmüller space of X into universal Teichmüller space is contracting. One of the elements of the proof involves analyzing the flex of certain points on the boundary of the unit ball in the Banach space Q(Y). The notion of flex is a sort of local version of the idea of reflexivity of Banach spaces. Suppose \(\Theta (\psi)=\phi\). To prevent loss of mass, the phase of \(\psi\) must nearly agree with that of the pull back of \(\phi\) to Y, at least over a region \(Y_ 0\) which contains most of the mass of \(| \psi |\). From the lemma on flex, one shows that agreement of phase implies the mass distribution of \(\psi\) mimics that of \(\phi\), which in the large is determined by the combinatorics of the covering \(Y\to X\). For a nonamenable covering, most of the mass of \(Y_ 0\) will be near its boundary, where the pairing is inefficient by a definite amount; this forces \(\| \Theta \| <1.\)

The arguments are carried further to estimate the dependence of \(\| \Theta \|\) on moduli.

Theorem. Let X be a hyperbolic Riemann surface of finite type, \(Y\to X\) an infinite-sheeted covering space with finitely generated fundamental group. Then \(\| \Theta_{Y/X}\| <c(n,L)<1\), where c(n,L) is a function of n the number of generators of \(\pi_ 1(Y)\) and L the length of the shortest geodesic on X and c(n,L) depends continuously on L. By using this result coupled with the theory of geometric limits of quadratic differentials, the author gives a new, analytic proof of the existence of a fixed point for Thurston’s “skinning” map. The skinning map is a key tool in Thurston’s construction of hyperbolic structures on 3-manifolds.

Now suppose f: \(Y\to X\) is a covering map. Then there is a natural operator \(\Theta\) : Q(Y)\(\to Q(X)\) defined by pulling back over the branches of \(f^{-1}\) and summing over all of the branches. This is the classical Poincaré theta operator when f is the universal covering of the unit disk onto a Riemann surface X. As an operator between the Banach spaces Q(Y) and Q(X), the norm of \(\Theta\) is less than or equal to 1. The chief result of this paper describes the types of coverings f for which the norm of \(\Theta\) is strictly less than 1. Say a covering f is amenable if there are large balls with small boundary in a graphic caricature of the covering. If X is hyperbolic and of finite type, then its universal cover is nonamenable.

Theorem. Let \(Y\to X\) be a covering of a hyperbolic Riemann surface X. Either:

1. The covering is amenable, and \(\Theta (B_ Y)=B_ Y\), or

2. The covering is nonamenable, and the closure of \(\Theta (B_ Y)\) is contained in the interior of \(B_ X.\)

As a corollary, if G is a nonabelian Fuchsian group acting on the unit disk \(\Delta\) such that \(X=\Delta /G\) is a compact Riemann surface of finite type, then the norm of \(\Theta\) is less than 1 and the inclusion mapping from the Teichmüller space of X into universal Teichmüller space is contracting. One of the elements of the proof involves analyzing the flex of certain points on the boundary of the unit ball in the Banach space Q(Y). The notion of flex is a sort of local version of the idea of reflexivity of Banach spaces. Suppose \(\Theta (\psi)=\phi\). To prevent loss of mass, the phase of \(\psi\) must nearly agree with that of the pull back of \(\phi\) to Y, at least over a region \(Y_ 0\) which contains most of the mass of \(| \psi |\). From the lemma on flex, one shows that agreement of phase implies the mass distribution of \(\psi\) mimics that of \(\phi\), which in the large is determined by the combinatorics of the covering \(Y\to X\). For a nonamenable covering, most of the mass of \(Y_ 0\) will be near its boundary, where the pairing is inefficient by a definite amount; this forces \(\| \Theta \| <1.\)

The arguments are carried further to estimate the dependence of \(\| \Theta \|\) on moduli.

Theorem. Let X be a hyperbolic Riemann surface of finite type, \(Y\to X\) an infinite-sheeted covering space with finitely generated fundamental group. Then \(\| \Theta_{Y/X}\| <c(n,L)<1\), where c(n,L) is a function of n the number of generators of \(\pi_ 1(Y)\) and L the length of the shortest geodesic on X and c(n,L) depends continuously on L. By using this result coupled with the theory of geometric limits of quadratic differentials, the author gives a new, analytic proof of the existence of a fixed point for Thurston’s “skinning” map. The skinning map is a key tool in Thurston’s construction of hyperbolic structures on 3-manifolds.

Reviewer: F.P.Gardiner

### MSC:

30C70 | Extremal problems for conformal and quasiconformal mappings, variational methods |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

30C62 | Quasiconformal mappings in the complex plane |

11F12 | Automorphic forms, one variable |

### Keywords:

expansion constant; hyperbolic 3-manifolds; quadratic differentials; Poincaré theta operator; amenable; Teichmüller space; flex### References:

[1] | [Ba] Baily, W.: Introductory lectures on automorphic forms. Princeton University Press, 1973 · Zbl 0256.32001 |

[2] | [Bers] Bers, L.: Spaces of degenerating Riemann surfaces. In: Discontinuous groups and Riemann surfaces. Ann. Math. Stud.76, 43–55 (1974) · Zbl 0294.32016 |

[3] | [Br] Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv.56, 581–596 (1985) · Zbl 0495.58029 · doi:10.1007/BF02566228 |

[4] | [Cha] Chabauty, C.: Limites d’ensembles et géometrie des nombers. Bull. Soc. Math. France78, 143–151 (1950) · Zbl 0039.04101 |

[5] | [DM] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math., Inst. Hautes Etud. Sci.36, 75–110 (1969) · Zbl 0181.48803 · doi:10.1007/BF02684599 |

[6] | [Earle] Earle, C.J.: Some remarks on Poincaré series. Compos. Math.21, 167–176 (1969) · Zbl 0194.39202 |

[7] | [EM] Earle, C.J., Marden, A.: Geometric complex coordinates for Teichmüller space. In preparation · Zbl 0678.32014 |

[8] | [Gabet] Gardiner, F.: Teichmüller theory and quadratic differentials. New York: Wiley Interscience 1987 · Zbl 0629.30002 |

[9] | [Gre] Greenleaf, F.P.: Invariant means on topological groups. Van Nostrand 1969 · Zbl 0174.19001 |

[10] | [Grom] Gromov, M.: Structures métriques pour les variétés riemanniennes. CEDIC, Textes mathématiques, 1981 |

[11] | [Habet] Harvey, W.J.: Spaces of discrete groups. In: Discrete groups and automorphic forms. New York: Academic Press 1977 |

[12] | [Kra 1] Kra, I.: Automorphic forms and Kleinian groups. New York: W.A. Benjamin, Inc 1972 · Zbl 0253.30015 |

[13] | [Kra 2] Kra, I.: Canonical mappings between Teichmüller spaces. Bull. AMS4, 143–179 (1981) · Zbl 0457.32011 · doi:10.1090/S0273-0979-1981-14870-9 |

[14] | [LS] Lyons, T., Sullivan, D.: Function theory, random paths and covering spaces. J. Differ. Geom.19, 299–323 (1984) · Zbl 0554.58022 |

[15] | [Masur] Masur, H.: The extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J43, 623–635 (1976) · Zbl 0358.32017 · doi:10.1215/S0012-7094-76-04350-7 |

[16] | [Mc] McMullen, C.: Iteration on Teichmüller space. Preprint · Zbl 0695.57012 |

[17] | [mil] Milman, D.P.: On some criteria for the regularity of spaces of type (B). Dokl. Akad. Nauk SSSR (N.S.)20, 243–246 (1938) · Zbl 0019.41601 |

[18] | [Oh] Ohtake, H.: Lifts of extremal quasiconformal mappings of arbitrary Riemann surfaces. J. Math. Kyoto Univ.22, 191–200 (1982) · Zbl 0497.30017 |

[19] | [Pet] Pettis, B.J.: A proof that every uniformly convex space is reflexive. Duke Math. J.5, 249–253 (1939) · Zbl 0021.32601 · doi:10.1215/S0012-7094-39-00522-3 |

[20] | [Pier] Pier, J.P.: Amenable locally compact groups. New York: Wiley Interscience 1984 · Zbl 0597.43001 |

[21] | [Poin] Poincaré, H.: Mémoire sur les fonctions Fuchsiennes. Acta Math.1, 193–294 (1882/3) · JFM 15.0342.01 · doi:10.1007/BF02592135 |

[22] | [PS1] Parson, L.A., Sheingorn, M.: Bounding the norm of the Poincaré {\(\theta\)}-operator. In: Analytic Number Theory. Lecture Notes Math. Vol.899. Berlin-Heidelberg-New York: Springer 1981 · Zbl 0475.10026 |

[23] | [PS2] Parson, L.A., Sheingorn, M.: Exponential sums connected with Ramanujan’s function {\(\tau\)}(n). Mathematika29, 270–277 · Zbl 0512.10029 |

[24] | [Ros] Rosenblatt, J.M.: A generalization of Følner’s condition. Math. Scand.33, 153–170 (1973) · Zbl 0272.28014 |

[25] | [RS] Reich, E., Strebel, K.: Extremal quasiconformal mappings with given boundary values. In: Contributions to Analysis. New York: Academic Press 1974 · Zbl 0318.30022 |

[26] | [Scott] Scott, P.: Subgroups of surface groups are almost geometric. J. London Math. Soc.17, 555–565 (1978) · Zbl 0412.57006 · doi:10.1112/jlms/s2-17.3.555 |

[27] | [Str1] Strebel, K.: On lifts of extremal quasiconformal mappings. J. Anal. Math.31, 191–203 (1977) · Zbl 0349.30016 · doi:10.1007/BF02813303 |

[28] | [Str2] Strebel, K.: On quasiconformal mappings of open Riemann surfaces. Comment. Math. Helv.53, 301–321 (1978) · Zbl 0421.30017 · doi:10.1007/BF02566081 |

[29] | [Str3] Strebel, K.: Quadratic differentials. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0547.30001 |

[30] | [Str4] Strebel, K.: Extremal quasiconformal mappings. MSRI preprint 1986 |

[31] | [Th1] Thurston, W.P.: Hyperbolic structures on 3-manifolds IV: Construction of hyperbolic manifolds. In preparation |

[32] | [Th2] Thurston, W.P.: Geometry and topology of three-manifolds. Princeton lecture notes, 1979 |

[33] | [Wol] Wolpert, S.: The Fenchel-Nielsen deformation. Ann. Math.115, 501–528 (1982) · Zbl 0496.30039 · doi:10.2307/2007011 |

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