There are no exotic ladder surfaces
TTHERE ARE NO EXOTIC LADDER SURFACES
ARA BASMAJIAN AND NICHOLAS G. VLAMIS
Abstract.
It is an open problem to provide a characterization of quasiconfor-mally homogeneous Riemann surfaces. We show that given the current literature,this problem can be broken into four open cases with respect to the topology ofthe underlying surface. The main result is a characterization in one of the theseopen cases; in particular, we prove that every quasiconformally homogeneous lad-der surface is quasiconformally equivalent to a regular cover of a closed surface(or, in other words, there are no exotic ladder surfaces). Introduction
A Riemann surface X is K - quasiconformally homogeneous , or K -QCH, if given anytwo points x, y ∈ X there exists a K -quasiconformal homeomorphism f : X → X such that f ( x ) = y . We say a Riemann surface is quasiconformally homogeneous ,or QCH, if it is K -QCH for some K (note: this definition diverges from the lit-erature, where such a surface is usually referred to as uniformly quasiconformallyhomogeneous). For a survey of the work on QCH surfaces see [2].For example, the Riemann sphere, the unit disk, and any Riemann surface whoseuniversal cover is isomorphic to the complex plane are all 1-QCH, or conformallyhomogeneous . In fact, this is a complete characterization of conformally homoge-neous Riemann surfaces, which leads us to the problem for which this paper isconcerned: Characterize all QCH Riemann surfaces.
Given the characterization of 1-QCH Riemann surfaces above, all the remainingcases to consider are hyperbolic Riemann surfaces (i.e. Riemann surfaces whoseuniversal cover is isomorphic to the unit disk).The starting point for such a characterization comes from higher dimensions. Thenotion of being K -QCH readily extends to the setting of hyperbolic manifolds ofany dimension. In dimension at least three, it was shown in [3, Theorem 1.3] thata hyperbolic manifold is QCH if and only if it is a (geometric) regular cover of aclosed hyperbolic orbifold. Naturally, such a result relies on rigidity phenomenain higher dimensions that do not occur in dimension two; in particular, as beingQCH is invariant under quasiconformal deformations, it is not too difficult to find ahyperbolic QCH surface that does not regularly cover a closed hyperbolic orbifold(see [3, Lemma 5.1]). a r X i v : . [ m a t h . G T ] S e p ARA BASMAJIAN AND NICHOLAS G. VLAMIS
This leads one to wonder—maybe naively—if every hyperbolic QCH surface is qua-siconformally equivalent to a cover of a closed hyperbolic orbifold? Interestingly,this is not the case: in [4, Theorem 1.1] the existence of quasiconformally exotic
QCH surfaces (i.e. QCH surfaces that are not quasiconformally equivalent to reg-ular covers of closed orbifolds) is shown. However, all the exotic QCH surfacesconstructed in [4] are homeomorphic; in particular, they are homeomorphic to theone-ended infinite-genus surface (affectionately referred to as the Loch Ness monstersurface).Our first theorem establishes that all QCH surfaces (and, in particular, exotic QCHsurfaces) are topological regular covers of closed surfaces, or in other words, thereare no topologically exotic
QCH surfaces:
Theorem 5.1.
Every quasiconformally homogeneous Riemann surface topologicallycovers a closed surface.
Note that every closed Riemann surface is QCH (see [3, Proposition 2.4] for a bound),so in the characterization of all QCH surfaces it is only left to consider non-compactsurfaces. As a corollary to Theorem 5.1, we see that, up to homeomorphism, thereare only a finite of number cases to consider. In particular, combining Theorem 5.1with the classification of non-simply connected, infinite sheeted, regular covers ofclosed surfaces (Proposition 5.2 below), we have:
Corollary 1.1.
Up to homeomorphism, there are six non-compact QCH Riemannsurfaces, namely the plane, the annulus, the Cantor tree surface, the blooming Can-tor tree surface, the Loch Ness monster surface, and the ladder surface . As an immediate consequence of Corollary 1.1, we can strengthen a result of Kwakkel–Markovic [10, Proposition 2.6]:
Corollary 1.2.
A Riemann surface of positive, finite genus is quasiconformallyhomogeneous if and only if it is closed.
Consider the non-hyperbolic cases in Corollary 1.1: we know that (1) every Rie-mann surface homeomorphic to the plane is QCH and (2) that a Riemann surfacehomeomorphic to the annulus is QCH if and only if its universal cover is isomorphicto C (this follows from the discussion of 1-QCH surfaces, Theorem 5.1, and the factthat the fundamental group of a closed hyperbolic Riemann surface does not have acyclic normal subgroup—this also follows from [3, Theorem 1.1]). This leaves onlyfour topological cases to consider.In this article, we give a characterization in one of the four cases: the ladder surface,that is, the two-ended infinite-genus surface with no planar ends. In this case,our main theorem shows that there are no exotic QCH ladder surfaces, yielding acomplete classification of QCH ladder surfaces: This nomenclature is explained in Proposition 5.2
HERE ARE NO EXOTIC LADDER SURFACES 3
Theorem 4.1.
A hyperbolic ladder surface is quasiconformally homogeneous if andonly if it is quasiconformally equivalent to a regular cover of a closed hyperbolicsurface.
Given Theorem 4.1, it is natural to ask if the distance (in the Teichm¨uller metric) ofa K -QCH ladder surface from a regular cover can be explicitly bounded as a functionof K . With this in mind, all of our proofs are written with the goal of providingexplicit bounds in terms of K for all constants that appear; however, we are unableto do this in one location, namely the constant A appearing in Lemma 4.3. It wouldbe interesting to find such a bound.The first step in the proof of Theorem 4.1 is to choose a distance-minimizing geodesic(that is, a proper embedding of R minimizing the distance between any two of itspoints); however, to do so, we need to know such a geodesic exists. Our final theoremprovides a sufficient topological condition for such a geodesic to exist; in addition, weshow that there can be no topological condition that is both necessary and sufficientfor a distance-minimizing geodesic to exist. Theorem 3.2.
Every non-compact hyperbolic Riemann surface with at least twotopological ends contains a distance-minimizing geodesic. Moreover, if an orientable,non-compact topological surface has a unique end, then it admits complete hyperbolicstructures containing distance-minimizing geodesics as well as complete hyperbolicstructures that do not admit such geodesics.
Despite the narrative arc of the results above, in what follows, the proofs of thetheorems will appear in reverse order.
Acknowledgements.
The authors are grateful to Richard Canary and Hugo Par-lier for helpful conversations.This project began several years ago during a visit of the second author to thefirst (before they were at the same institution) that was funded by the GEARNetwork and so: the second author acknowledges support from U.S. National ScienceFoundation grants DMS 1107452, 1107263, 1107367 ”RNMS: GEometric structuresAnd Representation varieties” (the GEAR Network). During that time the secondauthor was a postdoc at the University of Michigan and supported in part by NSFRTG grant 1045119. The second author is currently supported in part by PSC-CUNY Award
Preliminaries
Every Riemann surface is Hausdorff, orientable, and second countable; hence wewill require these attributes of all topological surfaces in this note.
ARA BASMAJIAN AND NICHOLAS G. VLAMIS
Hyperbolic geometry.
We mention some facts in hyperbolic geometry thatwill be used in the sequel. For more detailed information, see [6, 9, 13, 5].A homeomorphism f : U → V between domains U and V in C is K -quasiconformalif 1 K Mod(A) ≤ Mod(f(A)) ≤ KMod(A)for any annulus A in U , where Mod(A) is the modulus of A , that is, the unique realpositive number M such that A is isomorphic to { z ∈ C : 1 < | z | < e πM } . To extendto Riemann surfaces, we say a homeomorphism f : X → Y is K -quasiconformal ifthe restriction to any chart is K -quasiconformal.A Riemann surface X is hyperbolic if its universal cover is isomorphic to the unitdisk D . We can then realize X as the quotient of D by the action of a Fuchsiangroup Γ.If we equip D with its unique Riemannian metric of constant curvature -1, thenΓ acts on D by isometries and this metric descends to a metric on X , which willgenerally denote by ρ . Given a closed geodesic γ in a hyperbolic Riemann surface X , we let (cid:96) X ( γ ) denote its length in ( X, ρ ).We first recall some basic geometric properties of quasiconformal maps. Beforedoing so, we require some notation. Given two compact subsets C and C in ametric space ( M, d ), let d ( C , C ) denote the distance between the two subsets, thatis, d ( C , C ) = min { d ( x, y ) : x ∈ C , y ∈ C } , and let H ( C , C ) denote the Hausdorff distance between C and C , that is, H ( C , C ) = max { sup x ∈ C inf y ∈ C d ( x, y ) , sup y ∈ C inf x ∈ C d ( y, x ) } . Finally, given two metric spaces ( M , d ) and ( M , d ), a surjection f : M → N isan ( A, B )-quasi-isometry if1
A d ( x, y ) − B ≤ d ( f ( x ) , f ( y )) ≤ Ad ( x, y ) + B for all x, y ∈ M . Lemma 2.1.
Let Z be a hyperbolic Riemann surface and let γ be a simple closedgeodesic in Z . If f : Z → Z is K -quasiconformal, then(i) K (cid:96) Z ( γ ) ≤ (cid:96) Z ( f ( γ )) ≤ K(cid:96) Z ( γ ) ,(ii) f is a ( K, K log 4) -quasi-isometry, and(iii) there exists a constant R depending only on K and the length of γ so that H ( f ( γ ) , δ ) < R , where δ is the geodesic homotopic to f ( γ ) . Throughout our arguments, we will require the use of the collar lemma:
HERE ARE NO EXOTIC LADDER SURFACES 5
Theorem 2.2.
Let X be a hyperbolic Riemann surface and let η : R → R be givenby η ( (cid:96) ) = arcsinh (cid:18) (cid:96)/ (cid:19) . If γ and γ are disjoint simple closed geodesics of length (cid:96) and (cid:96) , respectively,then the η ( (cid:96) i ) -neighborhood of γ i , that is, the set A η ( (cid:96) i ) ( γ i ) = { x ∈ X : ρ ( x, γ ) < η ( (cid:96) ) } is embedded in X and A η ( (cid:96) ) ( γ ) ∩ A η ( (cid:96) ) ( γ ) = ∅ . We end with a special property of compact hyperbolic surfaces with totally geodesicboundary—that is, a compact surface arising as the quotient of a countable inter-section of pairwise-disjoint closed half planes in D by the action of a Fuchsian group.In a hyperbolic surface with totally geodesic boundary, an orthogeodesic is geodesicarc whose end points meet the boundary of the surface orthogonally.A pants decomposition of a topological surface is a collection of pairwise-disjointsimple closed curves, called the cuffs , such that each complementary componentof their union is homeomorphic to a thrice-punctured sphere. Every orientabletopological surface with non-abelian fundamental group has a pants decompositionand, in a compact hyperbolic surface (possibly with boundary), there always existsa pants decomposition where the curves are of bounded length, with the bounddepending only on the topology of the surface and the length of its boundary. Theorem 2.3 (Bers pants decomposition theorem) . Given positive real numbers A and L , there exists a positive real number B —depending only on A and L —suchthat every compact hyperbolic surface with totally geodesic boundary whose boundarylength is less than L and whose area is less than A admits a pants decompositionwith cuff lengths bounded above by B . Topological ends.
The notion of an end of a topological space was introducedby Freudenthal and, in essence, encodes the topologically distinct “directions” ofgoing to infinity in a non-compact space.More formally, for a non-compact second-countable surface S , fix an exhaustion { K n } n ∈ N of S by compact sets so that for each n ∈ N , K n lies in the interior of K n +1 and such that each component of the complement of K n is unbounded. Wethen define a (topological) end of S to be a sequence e = { U n } n ∈ N , where U n is acomplementary component of K n and U n ⊃ U n +1 .The space of ends of S , denoted E ( S ), is the set of ends of S equipped with thetopology generated by sets of the form ˆ U n = { e ∈ E ( S ) : U n ∈ e } . It is an exerciseto check that, up to homeomorphism, the definition of E ( S ) given does not dependon the choice of compact exhaustion. We will say that an open subset V of S withcompact boundary is a neighborhood of an end e = { U n } n ∈ N if there exists N ∈ N such that U N ⊂ V . ARA BASMAJIAN AND NICHOLAS G. VLAMIS
We say an end e = { U n } n ∈ N is planar if, for some N ∈ N , U N is planar (i.e.homeomorphic to a subset of R ). We denote the set of non-planar ends by E np ( S ),which is a closed subset of E ( S ). Note that E np ( S ) is non-empty if and only if S hasinfinite genus. Theorem 2.4 (Classification of surfaces (see [12])) . Two orientable surfaces withoutboundary, S and S , of the same (possibly infinite) genus are homeomorphic if andonly if there is a homeomorphism E ( S ) → E ( S ) sending E np ( S ) onto E np ( S ) . distance-minimizing geodesics and rays In a hyperbolic surface X , a distance-minimizing geodesic is a unit-speed geodesiccurve α : R → X such that d X ( γ ( a ) , γ ( b )) = | b − a | for all a, b ∈ R . Recall that amap is proper if the inverse image of a compact set is compact. Lemma 3.1.
Every distance-minimizing geodesic in a hyperbolic Riemann surfaceis proper.Proof.
Let X be a hyperbolic Riemann surface and let α : R → X be a continuousnon-proper map. Then, there exists a compact set K such that α − ( K ) is closedand not compact; in particular, α − ( K ) is unbounded while K is bounded. Hence, α cannot be a distance-minimizing geodesic. (cid:3) As an easy consequence of Lemma 3.1, no compact hyperbolic Riemann surfacecan have a distance-minimizing geodesic. In Theorem 3.2, we give a topologicallysufficient condition for distance-minimizing geodesics to exist in a non-compact hy-perbolic Riemann surface; however, in addition, we see that there cannot be anecessary topological condition for the existence of such a geodesic.
Theorem 3.2.
Every non-compact hyperbolic Riemann surface with at least twotopological ends contains a distance-minimizing geodesic. Moreover, if an orientable,non-compact non-planar topological surface has a unique end, then it admits com-plete hyperbolic structures containing distance-minimizing geodesics as well as com-plete hyperbolic structures that do not admit such geodesics.
We split the proof into three lemmas covering the separate cases. Let us first considerthe multi-ended case.
Lemma 3.3.
Let X be a hyperbolic Riemann surface. If X has at least two topo-logical ends, then X contains a distance-minimizing geodesic with distinct ends.Proof. Let e and e be distinct topological ends of X and let η be a separating,simple, closed geodesic separating e and e . Label the two components of X (cid:114) η by U and U so that U i is a neighborhood of e i . For i ∈ { , } , choose a sequence { x in } n ∈ N in U i such that lim x in = e i . Let γ n : I n → X be the minimal-length unit-speed geodesic curve between x n and x n . Observe that γ n intersects η (exactly once) HERE ARE NO EXOTIC LADDER SURFACES 7 for each n ∈ N ; let t n ∈ I n such that γ n ( t n ) ∈ η . By the compactness of the lift of η to the unit tangent bundle of X , the sequence { γ (cid:48) n ( t n ) } n ∈ N accumulates; let v bean accumulation point of the sequence and let α be the corresponding unit-speedgeodesic in X . Note that the endpoints of α are e and e and hence they aredistinct.We claim α is a distance-minimizing geodesic: assume not and let w and z be pointson α such that there exists a distance-minimizing path δ of length strictly less thanthat of the segment of α connecting w and z . Let β denote the segment of α connecting w and z and let ∆ = (cid:96) ( β ) − (cid:96) ( δ ). Choose a positive real number (cid:15) so that (cid:15) < ∆ and such that the (cid:15) -neighborhood Q of β (that is, Q = { x ∈ X : ρ ( x, δ ) < (cid:15) } )is isometric to the (cid:15) -neighborhood of a geodesic segment in H of length (cid:96) ( β ). Notethat Q has two geodesic sides, one containing w and the other containing z .Now there exists N ∈ N such that γ N ∩ Q is connected with endpoints w N and z N on the same sides of Q as w and z , respectively. If δ w and δ z are the shortest curvesin Q connecting w N to w and z N to z , respectively, then, as δ w ∪ δ ∪ δ z is a pathconnecting w N and z N , it follows that (cid:96) ( δ w ) + (cid:96) ( δ ) + (cid:96) ( δ z ) > (cid:96) ( γ N ∩ Q ) > (cid:96) ( β ) , where the first inequality follows from the fact that γ N is distance minimizing andthe second follows from β being the orthogonal connecting the geodesic sides of Q .But, at the same time, we have (cid:96) ( δ w ) + (cid:96) ( δ ) + (cid:96) ( δ z ) < (cid:96) ( δ ) + (cid:15) < (cid:96) ( δ ) + ∆ = (cid:96) ( β ) . However, this is a contradiction as both inequalities cannot hold. (cid:3)
We now move to the one-ended case. For Lemma 3.4 and Lemma 3.5 below, we re-mind the reader that, up to homeomorphism, there is a unique one-ended, orientablesurface whose end is non-planar, namely, the Loch Ness monster surface.
Lemma 3.4. If S is a non-planar one-ended, orientable surface, then there exists ahyperbolic Riemann surface X homeomorphic to S that does not contain a distance-minimizing geodesic.Proof. There are two cases: either the end of S is planar or not. Since S haspositive genus, if the end of S is planar, we can choose a hyperbolic Riemannsurface X homeomorphic to S in which the end of S corresponds to a cusp on X . Let α : R → X be a continuous function. If α fails to be proper, then it is notdistance minimizing by Lemma 3.1, so we may assume that α is proper. In this case,the two unbounded components of the intersection of α with a cusp neighborhoodbecome arbitrarily close; hence, α cannot be a distance-minimizing geodesic.Now suppose that the end of S is non-planar. Let c be any separating, simpleclosed curve in S . We inductively build a sequence of disjoint, separating, simpleclosed curves { c n } n ∈ N by requiring that c n +1 separates c n from the end of S . Let X be a hyperbolic Riemann surface such that there exists L >
ARA BASMAJIAN AND NICHOLAS G. VLAMIS b ba acP b α Figure 1.
On the left, P is a hyperbolic right-angled pentagon andfour copies of P are glued to form a square with a disk removed. Onthe right, these 1-holed squares are glued in a tiling extending in alldirections; the extension of the vertical geodesic α forms a propergeodesic arc.of the geodesic representative γ n of c n has length less than L . Now let α : R → X be a geodesic in X ; as before, we may assume that α is proper. By the lengthrestriction on the γ n , there exists some N ∈ N such that the intersection of α withthe bounded component of X (cid:114) γ N has length greater than L ; hence, α cannot bea distance-minimizing geodesic. (cid:3) Lemma 3.5. If S is a borderless one-ended, orientable surface, then there exists ahyperbolic Riemann surface X homeomorphic to S containing a distance-minimizinggeodesic.Proof. Again we split into two cases: first, suppose that the end of S is planar.In this case, let X be a hyperbolic Riemann surface homeomorphic to S such thatthe end of S corresponds to a funnel on X . All funnels have distance-minimizinggeodesics.Let us continue to the case where S has a non-planar end. Given a positive number b , there exists a unique right-angled hyperbolic pentagon P b having two consecutivesides of length b ; let the consecutive sides of P b have lengths b, b, a, c, a as in Figure 1.We can glue four copies of P b to form a 1-holed squared with outer boundary havinglength 8 b and inner boundary 4 c .We now build a bordered hyperbolic surface inductively: let T b be a copy of the1-holed square above. For n ∈ N , we construct T n +1 b by pasting eight copies of T nb to form a rectangle with 9 n − holes. We identify T nb with the middle copy of T nb in T n +1 b . We then let T b be the direct limit of the T nb . (Less formally, we obtain T b bytiling the plane with copies of the 1-holed square, see Figure 1.) HERE ARE NO EXOTIC LADDER SURFACES 9
Let R b be the hyperbolic Riemann surface obtained by identifying the bound-ary components of T b horizontally via an orientation-reversing isometry, so thata (dashed) circle is identified with the (solid) circle to its right in Figure 1. Notethat R b is infinite genus and one-ended and hence homeomorphic to S .Now let α be a vertical geodesic as in Figure 1. We claim that α is a distance-minimizing geodesic in R b . It is enough to prove that α minimizes distances betweenthe corners of 1-holed squares for which it passes. Let x and y be two such cornersand let γ a distance-minimizing path between them. By construction, the shortestpath from one side of a 1-holed square to any other is at least 2 b . It follows thatdistance between two infinite horizontal geodesics in T b is exactly 2 b . The same istrue in R b as the gluing of boundary components does not change height. Now if α crosses through n − x to y , then γ must do the sameand in particular the length of γ is at least 2 nb , which of course is the length of thesegment of α connecting x and y . (cid:3) Of course the difficulty in the one-ended case is that a proper arc needs to approachthe unique end of the surface in both the forwards and backwards directions. Tocapture this, we prove the existence of a distance-minimizing ray. Here, a ray is theimage of a continuous injective map of the half line [0 , ∞ ) ⊂ R . Proposition 3.6.
Every point on a non-compact hyperbolic Riemann surface is thebase point of some distance-minimizing ray.Proof.
We provide the sketch of the proof as the details are nearly identical to thosein the proof of Lemma 3.3. Let X be a non-compact hyperbolic Riemann surfaceand let e be a topological end of X . Fix a sequence { x n } n ∈ N that limits to e . Nowlet x be a point in X and, for n ∈ N , let γ n denote a unit-speed, minimal-length,geodesic curve starting at x and ending at x n . Let v n = γ (cid:48) n (0), then we may choosea unit vector v in the accumulation set of the sequence { v n } n ∈ N . Arguing as inLemma 3.3, the geodesic ray based at x determined by v is distance minimizing. (cid:3) QCH ladder surfaces are regular covers
In this section, we prove our main theorem:
Theorem 4.1.
A hyperbolic ladder surface is quasiconformally homogeneous if andonly if it is quasiconformally equivalent to a regular cover of a closed hyperbolicsurface.
It is not difficult to see that being QCH is a quasi-conformal invariant and thatevery regular cover of a closed hyperbolic surface is QCH (see [3, Proposition 2.7]);hence, to prove Theorem 4.1, we only need to focus on the forwards direction.The proof will be split into the lemmas in the subsections below. Throughout thesubsections below X denotes a K -QCH ladder surface. Let F K be the set of K -quasiconformal homeomorphisms X → X . We say that a simple closed curve in X separates the ends of X if its complement consists of two unbounded compo-nents.4.1. Shiga pants decomposition. A Shiga pants decomposition of a hyperbolicRiemann surface is a pants decompositions whose cuff lengths are uniformly boundedfrom above. The goal of this subsection is to show that every QCH ladder surfacehas a Shiga pants decomposition. It seems natural to expect a QCH surface to havesuch a pants decomposition, but, in fact, this is not always the case. For example,the surface R b constructed in the proof of Lemma 3.5 is QCH (it is a regular coverof a closed hyperbolic surface), but does not have a Shiga pants decomposition[11].The first step in the proof is to find a sequence of pairwise-disjoint simple closedgeodesics that separate the ends of X , that are of uniformly bounded length, andthat are “evenly” spaced throughout the surface (Lemma 4.2). We then show thatthe subsurfaces in the complement of these curves have bounded topology (thiswill follow from Lemma 4.3); the existence of a Shiga pants decomposition for X will follow by taking a Bers pants decomposition for each of these complementarysubsurfaces.Two real-valued functions f ( x ) and g ( x ) are said to be comparable , denoted f (cid:16) g ,if there exists positive constants A and B so that A ≤ f ( x ) g ( x ) ≤ B , for all x . Lemma 4.2.
There exists a sequence of pairwise-disjoint simple separating geodesics { γ n } n ∈ Z such that each γ n separates the ends of X and so that (1) ρ ( γ n , γ m ) (cid:16) H ( γ n , γ m ) (cid:16) | m − n | for all n, m ∈ Z . Moreover, the constants in the comparisons depend only on K and L = (cid:96) X ( γ ) .Proof. Choose any simple closed geodesic separating the ends of X and label it γ .Set L = (cid:96) X ( γ ). As X is two ended, by Lemma 3.3, we may choose a distance-minimizing geodesic β on X with distinct ends. Identify β with a unit-speed pa-rameterization β : R → X such that β (0) ∈ γ ; set x = β (0).Let R = R ( K ) be as in Lemma 2.1 and, for n ∈ Z , let x n = β (3 n ( R + KL )). As X is K -QCH, we may choose f n ∈ F K such that f n ( x ) = x n . Finally, let γ n be thegeodesic in the homotopy class of f n ( γ ). (Recall H ( γ n , f n ( γ )) ≤ R .)We claim that the sequence { γ n } n ∈ Z has the desired properties. To see this, firstobserve, for every n ∈ Z , that (cid:96) X ( γ n ) ≤ KL and γ n separates the ends of X . Next,we compute the distance between γ n and γ n +1 . Observe that we can construct apath from x n to x n +1 of length less than KL + R + ρ ( γ n , γ n +1 ) + R + KL , whichmust have length at least 3( R + KL ) as β is distance minimizing; hence, ρ ( γ n , γ n +1 ) ≥ R + KL ) − R − KL = R + 2 KL.
HERE ARE NO EXOTIC LADDER SURFACES 11
Regarding an upper bound, we have ρ ( γ n , γ n +1 ) ≤ ρ ( x n , x n +1 ) + 2 R = 3( R + KL ) + 2 R = 5 R + 3 KL, where the first inequality uses H ( f ( γ n ) , γ n ) < R .Assume m > n and recall that any path from γ n to γ m must pass through γ k for all n < k < m and hence(2) ρ ( γ n , γ m ) ≥ m − (cid:88) k = n ρ ( γ k , γ k +1 ) ≥ ( m − n )( R + 2 KL ) . It follows that ρ ( γ n , γ m ) >
0; in particular, γ n ∩ γ m = ∅ for all distinct n, m ∈ Z .Now, as γ n and γ m are disjoint, ρ ( γ n , γ m ) is realized by an orthogeodesic betweenthem. For the upper bound, using the fact that the orthogeodesic from γ n to γ m is shorter than the piecewise-continuous curve made up of orthogeodesics betweensuccessive γ k and arcs along the γ k we have(3) ρ ( γ n , γ m ) ≤ m − (cid:88) k = n (cid:20) ρ ( γ k , γ k +1 ) + (cid:96) X ( γ k )2 (cid:21) ≤ ( m − n ) (cid:20) (5 R + 3 KL ) + KL (cid:21) where the last inequality uses the fact that (cid:96) X ( γ k ) ≤ KL . Combining (2) and (3),we have shown(4) ( R + 2 KL ) ≤ ρ ( γ n , γ m ) | m − n | ≤ (5 R + 7 KL ρ ( γ n , γ m ) (cid:16) | n − m | .To show that ρ ( γ n , γ m ) is comparable to H ( γ n , γ m ), we first consider the followinginequality:(5) ρ ( γ n , γ m ) ≤ H ( γ n , γ m ) ≤ (cid:96) X ( γ n ) / ρ ( γ n , γ m ) + (cid:96) X ( γ m ) / ≤ KL + ρ ( γ n , γ m )Dividing the above inequality by ρ ( γ n , γ m ) we obtain(6) 1 ≤ H ( γ n , γ m ) ρ ( γ n , γ m ) ≤ KLρ ( γ n , γ m ) + 1For every integer n , using that γ n is the geodesic homotopic to f n ( γ ), we have that (cid:96) X ( γ n ) ≥ LK . Therefore, since γ n and γ m are disjoint for distinct integers n and m ,the collar lemma implies that the η (cid:0) LK (cid:1) -neighborhoods of γ n and γ m are embeddedand disjoint. In particular, we have ρ ( γ n , γ m ) ≥ η (cid:0) LK (cid:1) . Therefore, (6) becomes (7) 1 ≤ H ( γ n , γ m ) ρ ( γ n , γ m ) ≤ KL η ( LK ) + 1 . This finishes the proof of the lemma. (cid:3)
We remark that putting together lines (4) and (7) yields the concrete compari-son:(8) ( R + 2 KL ) ≤ H ( γ n , γ m ) | m − n | ≤ (cid:32) KL η ( LK ) + 1 (cid:33) (cid:18) R + 7 KL (cid:19) Lemma 4.3.
Let { γ n } n ∈ N be the sequence of geodesics constructed in Lemma 4.2and let Y n be the compact subsurface co-bounded by γ n and γ n +1 . There exists apositive real number A —depending on X —such that the area of Y n is at most A .Proof. Denote the lower bound of (4) by a and the upper bound of (8) by b . Let C = K log 4, then, as f ∈ F K is a ( K, C )-quasi-isometry,(9) ρ ( f ( γ n ) , f ( γ m )) ≥ K ρ ( γ n , γ m ) − C for all n, m ∈ Z .Choose m ∈ N satisfying m > Ka ( b + C + R ), where R is as in Lemma 2.1, andconsider the geodesic subsurface Z bounded by the geodesics γ − m and γ m . We set A to be the area of Z .Let f n ∈ F k be as defined as in the proof of Lemma 4.2. We claim that Y n ⊂ f n ( Z )for all n ∈ Z . Before proving our claim, note that the area of f n ( Z ) is boundedabove independent of n ∈ Z —the bound only depends on K and the area of Z . Tosee this, let Z n denote the geodesic straightening of f n ( Z ), so that the area of Z n agrees with that of Z . It must be that f n ( Z ) is in the R -neighborhood of Z n . Thearea of the R -neighborhood of Z n is bounded above by the area of Z n and R . As R only depends on K , we see that the area of f n ( Z ) is bounded as a function of K and the area of Z .Now to prove the claim, first note that using (9) we have:(10) ρ ( γ n , f n ( γ m )) ≥ ρ ( f n ( γ ) , f n ( γ m )) − R ≥ K ρ ( γ , γ m ) − C − R ≥ maK − C − R > b, where the last inequality follows from replacing m with the assumed lower boundin the choice of m .On the other hand, H ( γ n , γ n +1 ) < b and thus, ρ ( γ n , f n ( γ m )) > H ( γ n , γ n +1 ); in par-ticular, f n ( γ m ) must be disjoint from Y n . Observe that (10) holds with m replaced HERE ARE NO EXOTIC LADDER SURFACES 13 by − m ; hence, Y n and f n ( γ − m ) are disjoint. Thus Y n ⊂ f n ( Z ) and hence the areaof Y n is less than A . (cid:3) Remark 4.4.
Note that m can be explicitly chosen to be a function of K and L . A Bers pants decomposition of each Y n together with { γ n } n ∈ Z yields a pants de-composition for X with bounded cuff lengths, establishing: Proposition 4.5.
Every QCH ladder surface admits a Shiga pants decomposition. (cid:3)
Aside: coarse geometry.
We take a short tangent from the proof of Theorem 4.1to discuss the coarse geometry of QCH ladder surfaces.Note that as X is K -QCH, there is a lower bound on the injectivity radius of X depending only on K [3, Theorem 1.1]. In particular, the diameter of Y n isbounded above independent of n . Now let β be the distance-minimizing geodesicfrom Lemma 4.2. Define r : X → β by sending a point x of X to any point y ∈ β satisfying ρ ( x, y ) = ρ ( x, β ). It then follows that r is a quasi-isometry and hence β is a quasi-retract of X . Note that in the proof of Lemma 4.2, we could have chosenany distance-minimizing curve, establishing: Proposition 4.6. If X is a QCH ladder surface, then every distance-minimizinggeodesic in X is a quasi-retract of X . (cid:3) Corollary 4.7.
Every QCH ladder surface is quasi-isometric to R equipped withthe standard Euclidean metric. (cid:3) Let us show that Proposition 4.5 and Proposition 4.6 do not characterize the prop-erty of being QCH amongst hyperbolic ladder surfaces.First, we note again that there exist QCH surfaces that do not admit a Shiga pantsdecomposition.
Proposition 4.8.
There exists a hyperbolic ladder surface R and distance-minimizinggeodesic β in R such that β is a quasi-retract of R , R admits a Shiga pants decom-position, and R is not QCH.Proof. Let S be a topological ladder surface and let { a n , b n , c n } n ∈ Z be a pants decom-position for S as in Figure 2. Let R a hyperbolic surface such that (cid:96) ( a n ) = (cid:96) ( b n ) = 1and (cid:96) ( c n ) = (cid:12)(cid:12) n (cid:12)(cid:12) for all n ∈ Z , and such that the geodesic arc in the homotopy classof β is distance minimizing.By construction, R has a Shiga pants decomposition and moreover the nearest pointprojection r : R → β is a quasi-isometry. However, the injectivity radius of R goesto 0 and hence R cannot be QCH. (cid:3) Proposition 4.9.
There exists a hyperbolic ladder surface with a Shiga pants de-composition that is not quasi-isometric to Z (and hence not QCH). c a b c a c a c − a − c − a − c b b b − b − β Figure 2.
A pants decomposition for a (topological) ladder surface.
Proof.
Let Γ be the two-ended graph shown here:For i ∈ { , , } , let V i be a hyperbolic i -holed torus with each boundary componentof length 1. Let R be a hyperbolic surface obtained from Γ by taking a copy of V i for each valence i vertex and identifying boundary components according to theedge relations in Γ. The resulting surface R is quasi-isometric to Γ; in particular,it is a ladder surface with a Shiga pants decomposition. However, Γ and hence R isnot quasi-isometric to Z . (cid:3) These propositions lead us the follow question, which we end the aside with:
Question 4.10.
If a hyperbolic ladder surface has positive injectivity radius and isquasi-isometric to R , then is it QCH? Preferred Shiga pants decomposition.
In the previous section, we showedthat X admits a Shiga pants decomposition; however, this is just an existence state-ment and does not give us enough information to directly construct the desiredcovering map. The goal of this subsection is to modify the Shiga pants decompo-sition from Proposition 4.5 into a (topological) form we can use to build a decktransformation (the desired form is shown in Figure 2).It is not difficult to show the existence of the desired pants decomposition using acontinuity and compactness argument in moduli space; however, it is not possibleto extract explicit length bounds from such an argument. With a little extra ef-fort, we proceed in a fashion allowing for effective constants—this is the content ofLemma 4.11.Let Σ be a compact surface with non-abelian fundamental group. The pants graph associated to Σ, written P (Σ), is the graph whose vertices correspond to pantsdecompositions of Σ (up to isotopy) and where two vertices are adjacent if they HERE ARE NO EXOTIC LADDER SURFACES 15 abb (cid:48)
Figure 3.
The elementary move up to homeomorphism in a once-punctured torus switching b and b (cid:48) b b (cid:48) b b (cid:48) Figure 4.
The two possible types of elementary moves up to home-omorphism in a 4-holed sphere.differ by an elementary move. An elementary move corresponds to removing asingle curve α from the pants decomposition and replacing α with a curve that isdisjoint from all remaining curves of the pants decomposition and intersecting α minimally (see Figures 3 and 4 ).Define an equivalence relation ∼ on the vertices of P (Σ) by setting two pants de-compositions to be equivalent if they differ by a homeomorphism of Σ. The modularpants graph , written MP (Σ), is the graph whose vertices correspond to equivalenceclasses of pants decompositions of Σ; two vertices are connected by an edge if theyhave representatives in P (Σ) that are adjacent.As P (Σ) is connected [7], we have that MP (Σ) is connected; moreover, MP (Σ)has finitely many vertices and hence finite diameter (in the graph metric). Observethat, up to homeomorphism, there are at most two ways to replace a single curve ina given pants decomposition; in particular, none of the edges in the modular pantsgraph correspond to the elementary move shown in Figure 3.Let σ be a hyperbolic metric on Σ. Given a vertex v ∈ MP (Σ), define M σ ( v ) = min { M : there exists { c , . . . , c ξ } ∈ P (Σ) such that[ { c , . . . , c ξ } ] = v and (cid:96) σ ( c i ) ≤ M for all i ∈ { , . . . , ξ }} , where ξ = ξ (Σ) = 3 g − b is the topological complexity of Σ ( g is the genus of Σand b the number of boundary components of Σ). Lemma 4.11.
Let Σ be a compact surface possibly with boundary and with ξ (Σ) > .Let σ be a hyperbolic metric on Σ with injectivity radius m , let v ∈ MP (Σ) , and let M ∈ R such that M σ ( v ) ≤ M . Then, for all w ∈ MP (Σ) , M σ ( w ) is bounded aboveby a function of ξ, m, and M .Proof. Let P be a pair of pants representing v with cuff lengths all bounded aboveby M . If (cid:96) is the length of any orthogeodesic connecting a boundary component toitself, then sinh( (cid:96)/ ≤ cosh( M/ m/ v = { c , . . . , c ξ } for v such that (cid:96) σ ( c i ) ≤ M for all i ∈ { c , . . . , c ξ } . Let w ∈ MP (Σ) be adjacent to v . Up torelabelling, we can assume that there exists a representative ˜ w = { c (cid:48) , c , . . . , c ξ } of w adjacent to ˜ v .Let R denote the 4-holed sphere component of Σ (cid:114) (cid:83) ξi =2 c i and let P and P be thetwo pairs of pants in R sharing c as a common boundary component. Let α and α be the orthogeodesics in P and P , respectively, connecting c to itself. Up toDehn twisting about c , there exists a curve homotopic to c (cid:48) obtained by the takingthe union of α , α , and two subarcs of c . It follows that M σ ( w ) ≤ M + arccosh (cid:18) cosh(M / / (cid:19) . The result now follows by the fact that MP (Σ) is connected with finite diameter,which only depends on ξ . (cid:3) Let S be a ladder surface and fix a pants decomposition P = { a k , b k , c k } k ∈ Z as inFigure 2. Lemma 4.12. If X is a QCH ladder surface, then there exists a homeomorphism f : S → X such that f ( P ) is a Shiga pants decomposition for X .Proof. Let { γ n } n ∈ Z be the collection of curves guaranteed by Lemma 4.2. ByLemma 4.3, the complexity of the surface bounded by γ n and γ n +1 , denoted Y n ,is bounded. By Proposition 4.5, this guarantees the existence of a Shiga pantsdecomposition for X containing the collection of curves { γ n } ; let M be an upperbound for the lengths of the cuffs in this decomposition. Before continuing, we recallthat there is a lower bound on the injectivity radius of any K -QCH surface, whichonly depends on K [3, Theorem 1.1]; so, let K > X is K -QCH and let m = m ( K ) denote this lower bound.Fix a homeomorphism f : S → X such that for every n ∈ Z there exists k n ∈ Z with f ( c k n ) = γ n . Then, P n = f ( P ) ∩ Y n is a pants decomposition of Y n . As the Y n HERE ARE NO EXOTIC LADDER SURFACES 17 c a b c a c a c − a − c − a − c b b b − b − d d d − d − Figure 5.
The pants decomposition P along with the seams { d k } k ∈ Z determine Fenchel-Nielsen coordinates for hyperbolic structures on S .have bounded area and hence bounded topological complexity, ξ = max { ξ ( Y n ) : n ∈ Z } , is finite. Therefore, by Lemma 4.11, M Y n ([ P n ]) is bounded above by a function of ξ, m, and M . In particular, by pre-composing f with a homeomorphism of S , wemay assume that f ( P ) is a Shiga pants decomposition for X . (cid:3) Proof of Theorem 4.1.
We can now prove every QCH ladder surface is qua-siconformally equivalent to a regular cover a closed surface, finishing the proof ofTheorem 4.1. Before giving the proof, we recall a definition from Teichm¨uller the-ory.Let S be a (topological) ladder surface and let P = { a k , b k , c k } k ∈ Z be the pantsdecomposition of S given in Figure 2. Given a hyperbolic surface Z and a homeo-morphism h : S → Z , define the Fenchel–Nielsen coordinates of the marked surface( S, h ) be the collection of sextupletsFN((
S, h )) = { [ (cid:96) Z ( h ( a k )) , θ a k ( h ) , (cid:96) Z ( h ( b k )) , θ b k ( h ) , (cid:96) Z ( h ( c k )) , θ c k ( h )] } k ∈ Z , where θ a k , θ b k , and θ c k are the twist parameters associated to the curves a k , b k , and c k , respectively, with respect to a collection of seams { d k } k ∈ Z as in Figure 5. Thetwist parameters are given as an angle (as opposed to an arc length). Proof of Theorem 4.1.
Let X be a QCH ladder surface and let f : S → X be thehomeomorphism given by Lemma 4.12, so that f ( P ) is a Shiga pants decompositionfor X . By precomposing f with a (possibly infinite) product of Dehn twists aboutthe cuffs of P , we may assume that the twist parameters for X = ( S, f ) are between0 and 2 π .Now fix the marked surface ( S, h : S → Z ) such thatFN(( S, h )) = { [1 , , , , , } k ∈ Z . Since the cuff lengths and twist parameters of f ( P ) are bounded from above andbelow, we can conclude that the map h ◦ f − : X → Z is quasiconformal [1, Theo-rem 8.10]. To finish, we show that Z is a regular cover of a closed genus-3 hyperbolic surface:let τ : S → S be the horizontal translation determined, up to isotopy, by requiringthat τ (( a k , b k , c k , d k )) = ( a k +2 , b k +2 , c k +2 , d k +2 )for all k ∈ Z . Observe thatFN(( S, h ◦ τ − )) = FN(( S, h ))and hence τ h = h ◦ τ ◦ h − : Z → Z is isotopic to an isometry of Z . It follows that (cid:104) τ h (cid:105)\ Z is a closed hyperbolic genus-3 surface. (cid:3) Topology of QCH surfaces
The goal of this section is to prove that there are no topologically exotic QCHsurfaces, that is:
Theorem 5.1.
Every QCH surface is homeomorphic to a regular cover of a closedsurface.
Before proving Theorem 5.1, we need to understand the topology of a regular coverof a closed surface. The proposition below is stronger than we require, but withlittle extra work we state a more complete picture. Also, recall that, for the purposeof this article, surfaces are orientable and second countable.
Proposition 5.2.
A regular cover of a closed surface is either compact or homeo-morphic to one of the following six surfaces:(1) R ,(2) R (cid:114) ,(3) the Cantor tree surface, i.e. the planar surface whose space of ends is aCantor space,(4) the blooming Cantor tree surface, i.e. the infinite-genus surface with no pla-nar ends and whose space of ends is a Cantor space,(5) the Loch Ness Monster surface, i.e. the one-ended infinite-genus surface, or(6) the ladder surface, i.e. the two-ended infinite-genus surface with no planarends.Moreover, the torus is the only closed surface regularly covered by R (cid:114) .Proof. Let B be a closed surface and let π : S → B be a regular cover. It is notdifficult to show that the end space of a regular cover of a closed manifold is eitherempty, discrete with 1 or 2 points, or a Cantor space (this is a classical theorem ofHopf [8]). If the end space of S is empty, then S is compact; in the other cases, S is non-compact. If S is non-planar, then the co-compactness of the action of thedeck group associated to π on S will guarantee that every end of S is non-planar. HERE ARE NO EXOTIC LADDER SURFACES 19
Therefore, either S is compact; S is planar with 1, 2, or a Cantor space of ends; or S is infinite genus with 1, 2, or a Cantor space of ends, all of which are non-planar.Using the classification of surfaces, we see that—up to homeomorphism—there areonly six non-compact surfaces that meet these criteria, namely the ones listed above.We leave it as an exercise to show that, with the exception of R (cid:114) , each of thelisted surfaces covers a closed genus-2 surface.Finally, R (cid:114) is a regular cover of the torus; moreover, no surface of genus atleast two can be regularly covered by R (cid:114) : indeed, π ( R ) (cid:114) is cyclic and thefundamental group of a hyperbolic surface cannot have a normal cyclic subgroup. (cid:3) We can now prove Theorem 5.1.
Proof of Theorem 5.1.
Let X be a K -QCH surface. Further, for the sake of arguingby contradiction, assume that X is not a regular cover of a closed surface. If X is closed, then it is trivially a regular cover of a closed surface, namely itself. So,we may assume that X is non-compact. Note that under these assumptions, X isnecessarily hyperbolic.First, assume that X has positive (possibly infinite) genus and has at least oneplanar end. We can then choose a non-planar compact subsurface Y of X and anunbounded planar subsurface U of X such that ∂U is compact. Let { x n } n ∈ N bea sequence in U such that every compact subset of X contains only finitely manyof the x n . Fix x ∈ Y and let f n : X → X be a K -quasiconformal map such that f n ( x ) = x n . Note that as f n is a ( K, K log 4)-quasi-isometry (see Lemma 2.1),the diameter of f n ( Y ) is bounded as a function of K and the diameter of Y . Inparticular, as ρ ( x n , ∂Y ) → ∞ as n → ∞ , it must be that f n ( Y ) ⊂ U for large n ,but this is impossible as every subsurface of a planar surface is planar.We can now conclude that X is either (i) planar or (ii) has infinite genus and noplanar ends. In either case, as we are assuming X is not a regular cover of a closedsurface, by Proposition 5.2, X has at least three ends, one of which is isolated, callit e . (Note: if the end space does not contain an isolated point, then it is neces-sarily a Cantor space.) Let P be a compact subsurface in X with three boundarycomponents, each of which is separating, and such that each component of X (cid:114) P is unbounded and such that there exists a component U of X (cid:114) P with U a neigh-borhood of e . The argument now proceeds nearly identically to the previous case.Let { x n } n ∈ N be a sequence in U such that every compact subset of X contains onlyfinitely many of the x n . Fix x ∈ P and let f n : X → X be a K -quasiconformalmap such that f n ( x ) = x n . Again, as f n is a ( K, K log 4)-quasi-isometry, the diam-eter of f n ( P ) is bounded as a function of K and the diameter of P . In particular, ρ ( x n , P ) → ∞ as n → ∞ , it must be that f n ( P ) ⊂ U for large n , but this isimpossible as it would require X (cid:114) f n ( P ) to have a bounded component. (cid:3) References [1] Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su, and Zongliang Sun. OnFenchel-Nielsen coordinates on Teichm¨uller spaces of surfaces of infinite type.
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