MAPPINGS, TEICHMULLER QUASICONFORMAL HOMOGENEITY, AND NON-AMENABLE COVERS OF RIEMANN SURFACES
PETRA BONFERT-TAYLOR, GAVEN MARTIN, ALAN W. REID, AND EDWARD C. TAYLOR
Abstract. We show that there exists a universal constant Kc so that every K-strongly quasiconformally homogeneous hyperbolic surface X (not equal to H2 ) has the property that K > Kc > 1. The constant Kc is the best possible, and is computed in terms of the diameter of the (2, 3, 7)-hyperbolic orbifold (which is the hyperbolic orbifold of smallest area.) We further show that the minimum strong homogeneity constant of a hyperbolic surface without conformal automorphisms decreases if one passes to a non-amenable regular cover.
To F.W. Gehring - friend, mentor, and inadvertent match-maker1
1. Introduction and Statement of Results Recall that an orientable hyperbolic n-manifold N is uniformly quasiconformally homogeneous if there exists a constant K 1 so that for any two points x, y N there exists a K-quasiconformal automorphism of N that pairs x and y. The concept of quasiconformal homogeneity was introduced and developed by Gehring and Palka in [7]; for other work on quasiconformally homogeneous structures see [8], [9], [4], and [5]. In dimensions three and above, owing to well-known quasiconformal rigidity phenomena, the property of being uniformally quasiconformally homogeneous is a topologically restrictive one, we recall Theorem 1.3 of [4]. Theorem 1.1. Let N be an orientable hyperbolic n-manifold, with n 3. Then N is uniformly quasiconformally homogeneous if and only if N is the regular cover of a closed hyperbolic orbifold. Because quasiconformal rigidity phenomena fail in dimension two such a strong topological classication is unlikely to be true for surfaces. However by strengthening the denition of quasiconformal homogeneity, one can construct a setting from which interesting questions can be posed concerning the analytic properties of quasiconformal automorphisms of surfaces. A hyperbolic surface (equivalently a Riemann surface) X is strongly quasiconformally homogeneous ([5]) if there exists
The rst author was supported in part by NSF grant DMS-0305704. The second author was supported in part by the Marsden Fund, NZ. The third author was supported in part by NSF grant DMS-0503753. This author would also like to thank Wesleyan University for its hospitality during this work. The fourth author was supported in part by NSF grant DMS-0305704. 1 Two of the authors met and married while being postdoctoral students at the University of Michigan. Fred Gehring was instrumental in the events that made their meeting possible.
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2 PETRA BONFERT-TAYLOR, GAVEN MARTIN, ALAN W. REID, AND EDWARD C. TAYLOR
a constant K 1 so that for any two points x, y X there is a K-quasiconformal automorphism f : X X so that y = f (x) and so that f is homotopic to a conformal automorphism c : X X; we also say that X is K-strongly quasiconformally homogeneous. Using results of Gehring and Palka, it is elementary to see that every closed hyperbolic surface is strongly quasiconformally homogeneous, and thus this denition applies to a broad class of hyperbolic surfaces. We can provide a complete classication of strongly quasiconformally homogeneous surfaces, and in fact this is what motivates our interest in them. Using the argument used to prove Theorem 1.1, we observe: Theorem 1.2. Let X be an orientable hyperbolic surface. Then X is strongly quasiconformally homogeneous if and only if X is a regular cover of a closed hyperbolic orbifold. See the proof of Theorem 1.3 in [4]. Of course, there are many covers of a closed hyperbolic surface which are not regular covers and it is thus easy to construct an example of a hyperbolic surface X on which the injectivity radius function is bounded between two constants, and yet X is not uniformly quasiconformally homogeneous. Suppose that X is strongly quasiconformally homogeneous, and let Kaut (X) = inf{K | X is K-strongly quasiconformally homogeneous}. Using a normal family argument it is not hard to show that a strongly quasiconformally homogeneous hyperbolic surface X is in fact Kaut (X)-strongly quasiconformally homogeneous. We dene Kc = inf{Kaut (X) | X = H2 is strongly quasiconformally homogeneous}. One of our main results in this note is: Theorem 1.3. The constant Kc is strictly greater than one, and can be calculated in terms of the diameter of the quotient of H2 by the (2, 3, 7)-group. Furthermore no strongly quasiconformally homogeneous hyperbolic surface X = H2 is Kc -strongly quasiconformally homogeneous, that is, Kaut (X) is strictly greater than Kc . Remark. Each uniformly quasiconformally homogeneous hyperbolic surface X that is not H2 has the property that K(X) > 1. This follows from the fact that a 1-quasiconformal mapping is conformal, and from the fact that the conformal automorphism group of a non-elementary hyperbolic surface acts discontinuously on the surface (see Proposition 2.2 of [4].) In the proof of the following lemma we will need to localize our analysis. For a strongly quasiconformally homogeneous hyperbolic surface X we dene the function Kaut (x, y) = min {K(f )}
f
where the minimum is found over all quasiconformal mappings f : X X which are homotopic to a conformal automorphism and for which y = f (x). That the inmum is achieved is an elementary consequence of compactness properties of quasiconformal mappings. Now let X be a closed hyperbolic surface having trivial conformal automorphism group, and let Y be a regular cover of X. Though X has trivial automorphism group it remains strongly quasiconformally homogeneous since it is compact. We now establish the following useful lemma.
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Lemma 1.4. Let X be a closed hyperbolic surface having trivial automorphism group, and let Y be any regular cover of X. Then Kaut (Y ) Kaut (X). Proof. Because the surface X has trivial automorphism group, for each pair of points x1 , x2 X a best mapping realizing Kaut (x1 , x2 ) is homotopic to the identity and thus lifts to a family of Kaut (x1 , x2 )-quasiconformal automorphisms of Y , each homotopic to a conformal automorphism of Y , and so that for each such mapping there exist a pair of points y1 , y2 (respectively) in the bers 1 (x1 ) Y and 1 (x2 ) Y that is paired by the mapping. Thus one observes that Kaut (y1 , y2 ) Kaut (x1 , x2 ) for all y1 1 (x1 ) and y2 1 (x2 ). Because Y covers X regularly, the result follows. Remark. It is well known (e.g. see [6], section 3.2) that for genus g 3 the set of closed surfaces in Teichmller space having only trivial automorphism group is of u full measure. We show in [4] that if M n is a uniformly quasiconformally homogeneous hyperbolic manifold of any dimension n 2, then K(M n ) > 1 if and only if M n = Hn ; note that Hn is a non-amenable cover of any non-trivial hyperbolic manifold. In the setting of closed hyperbolic surfaces having trivial automorphism group we show that passing to any non-amenable regular cover strictly decreases Kaut : Theorem 1.5. Let X be a closed hyperbolic surface having trivial automorphism group, and let Y be a non-amenable regular cover of X. Then Kaut (Y ) < Kaut (X). 2. The Proof of Theorem 1.3 In this section we will prove Theorem 1.3. Before doing so, we will need to recall some basic denitions and facts. First, in the denition of strong quasiconformal homogeneity we are assuming that each allowable quasiconformal mapping is homotopic to a conformal mapping. Thus we can convert, by post-composition with the inverse of the conformal automorphism, each allowable quasiconformal mapping into one which is homotopic to the identity having the same dilatation. Since we wish to measure the dilation, the following function will be of use. Let : [0, ) [1, ) be the function which gives the best dilatation over all quasiconformal homeomorphisms of D2 that are homotopic to the identity and move the origin 0 a prescribed distance d [0, ), i.e. (d) = min{K 1 | there exists h : D2 D2 , Kqc, h|D2 = id, (0, h(0)) = d}. (Here, of course, (D2 , ) denotes the ball model of 2-dimensional hyperbolic space of constant curvature 1.) We record the following explicit formula for , due originally to Teichmller [13]. u Proposition 2.1. Let f : D2 D2 be a quasiconformal map which extends to the identity on the unit circle. Then K(f ) ((0, f (0))), where : [0, ) [1, ) is the increasing homeomorphism given by the function (d) = coth2 2 4(ed ) = coth2 1 e2d ,
and (r) is the modulus of the Grtsch ring whose complementary components are o 2 and [1/r, ] for 0 < r < 1. In particular, D (d) 16d2 as d 4 and (d) 1 + d as d 0. 2
4 PETRA BONFERT-TAYLOR, GAVEN MARTIN, ALAN W. REID, AND EDWARD C. TAYLOR
The critical value of d, for our analysis, is the minimum diameter of a hyperbolic orbifold (surface). In fact, the minimum diameter hyperbolic orbifold in dimension two is the minimum volume hyperbolic orbifold, that is, the orbifold built by the (2, 3, 7)-triangle group. The following fact is known, however we include a proof for convenience. Proposition 2.2. The minimum diameter hyperbolic orbifold Omin is the (2, 3, 7)hyperbolic orbifold. Proof. Recall the isodiametric inequality in hyperbolic 2-space: If a planar set has diameter d (d > 0) then the area of the planar set is less than or equal to 4 sinh2 ( d ) (e.g. see [12], also recall that 4 sinh2 ( d ) is the area of a hyperbolic 4 4 disk of radius d .) Using the convex polyhedron (say a Dirichlet polyhedron) of a 2 closed orbifold O2 of diameter d < , we easily see that the hyperbolic area of O2 is thus less than or equal to 4 sinh2 ( d ). 4 We now consider the (2, 3, 7)-triangle group. It is an easy exercise in hyperbolic trigonometry (e.g. see [2]) that the diameter of the orbifold quotient of H2 by the (2,3,7)-triangle group is approximately 0.62067. Using this value for d, we observe from the isodiametric inequality that any orbifold having diameter less than the diameter of the (2, 3, 7)-triangle orbifold must have area less than 0.305. Now using this area bound, we can systematically rule out the possibility that whole classes of Fuchsian groups have quotient orbifolds of diameter less than 0.62067. First note that it is clear that any Fuchsian group that contains a parabolic, or is of the second kind, or has innitely generated fundamental group, has a quotient surface of innite diameter and thus is not a candidate. From the area signature formula (see Theorem 10.4.3 in [2]), we immediately observe that if the genus of any such orbifold is greater than or equal to 1 then its area is greater than or equal to and thus is too large to have diameter less than 0.62067. In fact, any admissable Fuchsian group of genus 0 and with signature (0 : m1 , . . . , mr ) with r 4 will have area that is strictly greater than 0.305, and so these groups are ruled out as well. Thus we are left to consider groups of signature (0 : m1 , m2 , m3 ), where without loss of generality we can assume m1 m2 m3 . The basic idea is to observe that there is a monotonicity in the size of diameter in terms of the values of m1 , m2 and m3 . Using hyperbolic trigonometry one can rst explicitly show that if m1 = 2, then the diameter of the quotient of any admissable Fuchsian group of that signature is strictly greater than 0.62067. Now one considers admissible signatures for which m1 3. Once again, by explicit calculation using this monotonicity, we need only check a nite number of signatures and so we are able to rule these out as well. Remark. We conjecture that a minimum diameter hyperbolic 3-orbifold is the orientation-preserving half of the Z2 extension of the Coxeter 3-5-3 reection group. We restate Theorem 1.3 in terms of the discussion above. Theorem 2.3. We have Kc = (diam Omin ) 1.36138. Furthermore, any strongly quasiconformally homogeneous surfaces R = H2 satises Kaut (R) > Kc . Proof. We will rst construct a sequence {Sn } of surfaces such that the inmum over the sequence has the property that inf Kaut (Sn ) (diam Omin ). Then we
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will show that Kaut (S) > (diam Omin ) for each surface S, and this will complete the proof of the theorem. By a standard geometric application of residual niteness of Fuchsian groups (see [11]) we can construct a sequence of regular closed hyperbolic surface covers {Sn } of Omin , such that their minimal injectivity radii (Sn ) go to innity as n . Let x, y Sn , then there exists g Aut(Sn ) such that (g(x), y) diam Omin . Since the injectivity radius at the point g(x) is, by construction, necessarily big for all large n, we have that there exists a quasiconformal homeomorphism f : Sn Sn with f = id outside of B(g(x), injg(x) ), and f (g(x)) = y with K(f ) (diam Omin ) + n , where n gets smaller as the injectivity radius gets larger, and in the limit n 0 as n . To verify this, observe that the hyperbolic distance between g(x) and y in the hyperbolic metric of the disk B(g(x), injg(x) ) is only slightly larger than it is in Sn if injg(x) is large enough. We can thus transport Teichmllers extremal u map into this disk and use the identity map outside of the disk to map g(x) to y. Thus inf{Kaut (Sn ) | n N} (diam Omin ). Next we show that any strongly quasiconformally homogeneous surface = S H2 satises that Kaut (S) > Kc . Let S = H2 be an arbitrary strongly quasiconformally homogeneous surface, and choose x, y S such that min{(x, g(y)) | g Aut(S)} diam Omin . By composing with conformal automorphisms we can furthermore assume that a least dilatation mapping f that maps x to y while being homotopic to a conformal automorphism is in fact homotopic to the identity. We will show that K(f ) > Kc and this shows that Kaut (S) > Kc . Let p : D2 S be a universal covering map such that p(0) = x and p() = y, where (0, ) = (x, y) (and (0, 1). Let f be a lift of f to the unit disk such (0) = . Then the extension of f to the unit circle is the identity map of that f the unit circle. Hence, using Proposition 2.1, we have that K(f ) ((0, )) (diam(Omin )). We will show that the rst of these two inequalities is strict. First note that f cannot be Teichmllers unique minimal map (i.e. the unique u minimum dilatation quasiconformal mapping of the unit disk to itself that extends to the identity on the boundary and maps the origin to the point ). The key idea here is that Teichmllers extremal map cannot be compatible with any Fuchsian u group of the rst kind, and thus cannot live on any strongly quasiconformally homogeneous surface (compare Theorem 1.2). A general argument can be made using the fact that the unit disk is a nonamenable cover of the surface S (see Lemma 3.2), but we can give an explicit argument which only requires a geometric understanding of Teichmllers minimal u map ([13]). Let O be the double cover of the unit disk, branched at the origin. Let 1 : O D2 be the function 1 (z) = z which maps O \ {0} conformally onto the unit disk minus the origin. The two slits from 0 to on the two leafs of O get mapped onto the line segment [i , i ]. The unit disk minus this slit can be mapped conformally by a mapping 2 (via elliptic integrals) onto a round annulus A with
6 PETRA BONFERT-TAYLOR, GAVEN MARTIN, ALAN W. REID, AND EDWARD C. TAYLOR
inner radius 1 and outer radius R, here R depends only on . Furthermore, we can choose 2 such that 2 ([i , i ]) = D2 , and 2 is symmetric with respect to both the x and y-axis, in particular, 2 (i ) = i. Finally, let 3 (z) = z 1/z. This mapping maps A conformally onto the ellipse E1 with semi-axes R 1/R and R + 1/R and foci i, with a slit along the imaginary axis from 2i to 2i. Dene three more maps 1 , 2 , 3 , where 1 = 1 , 2 = 2 , but 3 is given by 3 (z) = z + 1/z. This mapping maps the annulus A onto a dierent ellipse: its semi axes are R + 1/R and R 1/R and its foci are the points 2. The composition = 3 2 1 is a conformal mapping from O \ [, 0] onto the ellipse E1 \[2i, 2i], and extends to the slits from 0 to in O so that (0) = 0, (1, 0) = (i(R + 1/R), i(R + 1/R)), and (0, 1) = ((R 1/R), R 1/R). Here, (1, 0) stands for the image under of the two lines on the double cover O of the unit disk above the negative real axis in D2 (and similarly interpreted for (0, 1) ). Similarly, the composition = 3 2 1 is a conformal mapping from O \ [, 0] onto the ellipse E2 \ [2, 2]. Note that can be extended to the slits, and the extension has a branch point at , i.e. extends to the double cover O of the unit disk, branched at . The mapping is nally obtained by mapping E1 onto E2 with the ane map T (u + iv) = u R+ R
1 R 1 R
+ iv
R R+
1 R 1 R
.
1 1 1 Then 1 2 3 T 3 2 1 is a quasiconformal mapping from O (branched at 0) onto O (branched at ) that agrees on both sheets and thus descends to a quasiconformal mapping : D2 D2 that maps 0 to . The only place where picks up quasiconformal dilatation is the mapping T that sends the ellipse E1 onto the ellipse E2 . We will now analyze the direction of maximal distortion for points z (1, 0) and points w (0, 1). The points z (1, 0) in the unit disk correspond to points on the imaginary axis in E1 , and points w (0, 1) correspond to points on the real axis in E1 . Thus innitesimal circles centered at points z (1, 0) get mapped under onto innitesimal ellipses centered at points on (1, ) with major axis orthogonal to R. On the other hand, innitesimal circles centered at points w (0, 1) get mapped under onto innitesimal ellipses centered at points on (, 1) with major axis along R. Let now be a Fuchsian group of the rst kind. Then contains a hyperbolic element whose axis A is arbitrarily close to (1, 1). Since is a smooth mapping on D2 \ {0}, the line eld of varies continuously in D2 \ {0}. Hence, on a segment of the axis A the line eld of is almost vertical, whereas on another segment of A the line eld of is almost horizontal. But some power of (or 1 ) maps points from the vertical segment into the horizontal segment, but the image of the vertical line eld under D is not the horizontal line eld, and thus = . Since is the identity on D2 , the only possibility for to be compatible with would be to satisfy = for all . Thus no Fuchsian group of the rst kind is compatible with , and so is not the lift of any quasiconformal mapping on any surface whose underlying Fuchsian group is of the rst kind. In particular, we have shown that our original mapping f (the lift of f : S S to the unit disk) cannot agree with the mapping . Since is unique with minimal
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distortion via Teichmllers result, we conclude that K(f ) > K() and this proves u the theorem. Remark. A fully general n-dimensional analogue (n 3) to the Teichmller u extremal result is not known. However, under certain restrictive assumptions a solution has been developed ([1]); it is shown that an extremal map is a rotation of the 2-dimensional extremal mapping. 3. Amenability and passage of Kaut to a cover Let X be a closed hyperbolic surface, and Y be a regular cover of X. Since a homeomorphism of X may not lift equivariantly to a homeomorphism of Y , it is hard to relate the quasiconformal homogeneities of X and Y . However, if we assume that X has trivial conformal automorphism group, then we can cite Lemma 1.4, and thus we quickly observe that K(Y ) K(X). If Y is a non-amenable regular cover then we can promote this inequality to a strict inequality. In order to introduce amenability we rst must x some notation. Let G be a graph and V be any set of vertices in G. The boundary V of V is the set of vertices in G V that are a distance one from V (that is, there is an edge in G that connects a vertex in V to a vertex in V .) Dene the expansion of G to be the inmum of |V|| as V varies over all nite vertex subsets of G. The group |V G is said to be amenable if = 0, and if G is not amenable it is non-amenable. Let Y be a regular cover of X, which we will denote by : Y X. We say that Y is an amenable regular cover if the covering group is amenable (here the graph in question is a graph of the group; the property of being amenable persists to every graph of a group.) If the covering group is non-amenable we say that Y is a non-amenable cover of X. See McMullen [10] for a more general presentation of amenable and non-amenable covers of Riemann surfaces. The following is our second primary result in this note. Theorem 3.1. Let X be a closed Riemann surface with Aut(X) = {id}. Let Y be a regular, non-amenable cover of X. Then Kaut (Y ) < Kaut (X). In order to prove this theorem we rst show a local version: Lemma 3.2. Suppose that X is a closed Riemann surface having trivial automorphism group, and Y be a regular non-amenable cover of X. Let x1 , x2 be two distinct points in X and let y1 1 (x1 ), y2 1 (x2 ) be preimages of x1 , x2 under the covering map . Then Kaut (y1 , y2 ) < Kaut (x1 , x2 ). Proof. Let x1 , x2 X be two distinct points. Let f be a quasiconformal mapping of X that is homotopic to the identity, that maps x1 to x2 and that satises K(f ) = Kaut (x1 , x2 ). Let X1 be the punctured surface X \ {x1 }, and let X2 the surface X \ {x2 }. Then f|X1 : X1 X2 is extremal in its homotopy class since otherwise there would be a m...