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• W4225665231 abstract "We show that Mirzakhani's curve counting theorem also holds if we replace surfaces by orbifolds. The goal of this note is to prove that this statement remains true when Γ $Gamma$ has torsion, that is, when O = H 2 / Γ $operatorname{O}=mathbb {H}^2/Gamma$ is an orbifold instead of a surface. Theorem 1.1.Let Γ ⊂ PSL 2 R $Gamma subset operatorname{PSL}_2mathbb {R}$ be a non-elementary finitely generated discrete subgroup and O = H 2 / Γ $operatorname{O}=mathbb {H}^2/Gamma$ the associated 2-dimensional hyperbolic orbifold. Then the limit As was already the case for the proof that we gave in [7] of Mirzakhani's (1.1), we will derive Theorem 1.1 from the weak-*-convergence of certain measures on the space C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ of currents, which is the space of π 1 or ( O ) $pi _1^{operatorname{or}}(operatorname{O})$ -invariant Radon measures on the set of geodesics on the orbifold universal cover O ∼ $tilde{operatorname{O}}$ of O $operatorname{O}$ . Trusting that the reader is familiar with currents, we just recall at this point that the set R ⩾ 0 S or ( O ) $mathbb {R}_{geqslant 0}mathcal {S}^{operatorname{or}}(operatorname{O})$ of weighted curves is a dense subset of C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ , that C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ is a cone in a linear space, and that the action of Map or ( O ) $operatorname{Map}^{operatorname{or}}(operatorname{O})$ on S or ( O ) $mathcal {S}^{operatorname{or}}(operatorname{O})$ extends to a linear action on C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ . We will recall a few facts about currents in Section 4.1 below, but we do already at this point refer the reader to [1-4, 7] for details and background. Theorem 1.2.Let O $operatorname{O}$ be a compact orientable non-exceptional hyperbolic orbifold with possibly empty totally geodesic boundary and let C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ be the associated space of geodesic currents. There is a Radon measure m Thu $mathfrak {m}_{operatorname{Thu}}$ on C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ such that for any γ 0 ∈ S or ( O ) $gamma _0in mathcal {S}^{operatorname{or}}(operatorname{O})$ , we have Theorem 1.3.Let O $operatorname{O}$ be a compact orientable hyperbolic orbifold with possibly empty totally geodesic boundary and let C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ be the associated space of geodesic currents. Then the limit It is known that, at least as stated, the limit (1.1) does not hold for non-orientable surfaces [9, 11] and this is why we assumed in the theorems above that the orbifold is orientable. It is, however, worth noting that all results here remain true for non-orientable orbifolds whose underlying topological space is an orientable surface. An example is D Σ / τ $DSigma /tau$ where D Σ $DSigma$ is the double of Σ $Sigma$ , an orientable surface with boundary, and τ $tau$ is the involution interchanging the copies of the surface. The reason why the theorems remain true is that, up to passing to finite index subgroups, the mapping class group of such an orbifold is isomorphic to the mapping class group of an orientable surface for which we know that the analog of Theorem 1.2 holds. Anyways, we decided against extending the theorems above to this kind of non-orientable orbifolds because (1) it would make the paper much harder to read and (2) we do not have any concrete applications in mind. In Section 2 we recall some facts and definitions about orbifolds, maps between orbifolds, the mapping class group of orbifolds, and such. In Section 3 we state precisely what we mean by as simple as possible and state, without proof, Proposition 3.3. In Section 4 we recall a few facts about currents and, assuming Proposition 3.3, prove Theorem 1.2 and the other results mentioned above. In Section 5 we prove a few facts needed in Section 6, where we prove Proposition 3.3. In this section, we recall a few basics about orbifolds such as definitions, (hyperbolic) orbifolds as orbit spaces, and mapping class groups. We also fix some notation that we will use throughout the paper. This is why we also encourage readers who already know all about orbifolds to at least skim over this section. The orbifold is orientable if all group actions Γ i ↷ U ̂ i $Gamma _icurvearrowright skew1hat{U}_i$ and all embeddings ϕ ̂ i , j $hat{phi }_{i,j}$ are orientation preserving. Similarly if we replace orientable by smooth. An orbifold with boundary is defined in the same way but this time the sets U ̂ i $skew1hat{U}_i$ are assumed to be open in R ⩽ 0 × R n − 1 $mathbb {R}_{leqslant 0}times mathbb {R}^{n-1}$ . An n $n$ -dimensional hyperbolic orbifold is one where the sets U ̂ i $skew1hat{U}_i$ are contained in H n $mathbb {H}^n$ , where the actions Γ i ↷ U ̂ i $Gamma _icurvearrowright skew1hat{U}_i$ preserve the hyperbolic metric, and where the maps ϕ ̂ i , j $hat{phi }_{i,j}$ are isometric embeddings. To define what is a hyperbolic orbifold with totally geodesic boundary, then one copies what we just wrote, only replacing H n $mathbb {H}^n$ by a closed half-space therein. As is the case in the world of manifolds, orbifolds have maximal orbifold atlases, orientable orbifolds have maximal orientable orbifold atlases, smooth orientable orbifolds have maximal smooth orientable orbifold atlases, and so on. We will always assume that our orbifolds (with adjectives) are equipped with maximal atlases (with adjectives). We refer to [18] for more on orbifolds. A point p $p$ in an orbifold O $operatorname{O}$ is singular if there are an orbifold chart ( U , U ̂ , Γ , ϕ ) $(U,skew1hat{U},Gamma ,phi )$ and p ̂ ∈ U ̂ $hat{p}in skew1hat{U}$ with ϕ ( p ̂ ) = p $phi (hat{p})=p$ and satisfying that Stab Γ ( p ̂ ) ≠ Id $operatorname{Stab}_Gamma (hat{p})ne operatorname{Id}$ . A point which is not singular is regular. We denote by sing ( O ) $operatorname{sing}(operatorname{O})$ the set of singular points of O $operatorname{O}$ . all orbifolds in this paper are such that sing ( O ) $operatorname{sing}(operatorname{O})$ is a proper subset of O $operatorname{O}$ . In the cases we are interested in, namely, compact orbifolds which are orientable, connected, and 2-dimensional, we have that sing ( O ) $operatorname{sing}(operatorname{O})$ is, in fact, a finite set of points in the interior of O $operatorname{O}$ . Remark.Whenever we need to choose a base point in our orbifold O $operatorname{O}$ , for example, when working with the fundamental group π 1 or ( O ) $pi _1^{operatorname{or}}(operatorname{O})$ , then we will assume without further mention that the base point is regular. The reader might amuse themselves by thinking about what the right notion of base point in the category of orbifolds would be if they allowed singular points to be base points. If O $operatorname{O}$ is a compact orbifold with boundary then we will say that a curve is essential if it is not freely homotopic into the boundary and if the associated free homotopy class is that of an infinite order element in the orbifold fundamental group. We will also denote by S or ( O ) $mathcal {S}^{operatorname{or}}(operatorname{O})$ the set of all free homotopy classes of essential curves in O $operatorname{O}$ . Following word-by-word the usual construction of the universal cover of a manifold but replacing homotopies by homotopies of orbifold maps, one gets the orbifold universal cover O ∼ $tilde{operatorname{O}}$ of the orbifold O $operatorname{O}$ . As is the case for manifolds, the fundamental group π or ( O ) $pi ^{operatorname{or}}(operatorname{O})$ acts discretely on the universal cover O ∼ $tilde{operatorname{O}}$ . Similarly, orbifold maps f : O → O ′ $f:operatorname{O}rightarrow operatorname{O}^{prime }$ between orbifolds induce homomorphisms f ∗ : π 1 or ( O ) → π 1 or ( O ′ ) $f_*:pi _1^{operatorname{or}}(operatorname{O})rightarrow pi _1^{operatorname{or}}(operatorname{O}^{prime })$ between the associated orbifold fundamental groups and lift to f ∗ $f_*$ -equivariant maps f ∼ : O ∼ → O ∼ ′ $tilde{f}:tilde{operatorname{O}}rightarrow tilde{operatorname{O}}^{prime }$ between the universal covers. Notation. Let O ∼ ⊂ H 2 $tilde{operatorname{O}}subset mathbb {H}^2$ be a closed connected (2-dimensional) subset of the hyperbolic plane with possibly empty geodesic boundary, let Γ ⊂ PSL 2 R $Gamma subset operatorname{PSL}_2mathbb {R}$ be a discrete subgroup which preserves O ∼ $tilde{operatorname{O}}$ and such that the induced action Γ ↷ O ∼ $Gamma curvearrowright tilde{operatorname{O}}$ is cocompact, and denote by In this setting, O ∼ $tilde{operatorname{O}}$ is the orbifold universal cover of O $operatorname{O}$ and Γ = π 1 or ( O ) $Gamma =pi _1^{operatorname{or}}(operatorname{O})$ is its orbifold fundamental group. It is not hard to see that the orbifolds O $operatorname{O}$ we are interested in are homeomorphic as topological spaces to surfaces, that is, to 2-dimensional manifolds. Such homeomorphisms do, however, destroy the orbifold structure. In fact, much more information is encoded in the surface that we get by deleting the singular points of O $operatorname{O}$ . Since we want to work with compact surfaces, we instead delete small balls around the singular points. Remark.We denote by B hyp ( q , r ) ⊂ O ∼ $B_{operatorname{hyp}}(q,r)subset tilde{operatorname{O}}$ the hyperbolic ball of radius r $r$ around q $q$ . Equivalently, Although our definition of the mapping class group differs from theirs (we do not have twists around the boundary), we refer to the book [8] by Farb and Margalit for background on the mapping class group. Lemma 2.1.Suppose that p ∈ sing ( Γ ) $pin operatorname{sing}(Gamma )$ and r > 2 δ $r>2delta$ are such that ρ $rho$ and ρ hyp $rho _{operatorname{hyp}}$ agree on B hyp ( p , r ) ∖ B hyp ( p , 2 δ ) $B_{operatorname{hyp}}(p,r)setminus B_{operatorname{hyp}}(p,2delta )$ , and let η ⊂ B hyp ( p , r ) ∖ B hyp ( p , δ ) $eta subset B_{operatorname{hyp}}(p,r)setminus B_{operatorname{hyp}}(p,delta )$ be a ρ $rho$ -geodesic segment whose boundary points are contained in ∂ B hyp ( p , r ) $partial B_{operatorname{hyp}}(p,r)$ . If η $eta$ is simple, then η $eta$ meets every ρ hyp $rho _{operatorname{hyp}}$ -geodesic ray emanating out of p $p$ at most once. In particular, η $eta$ has at most length 2 π sinh ( r ) $2pi sinh (r)$ . Note that Γ $Gamma$ -invariance of ρ $rho$ implies that it descends to a metric on Σ $Sigma$ which we once again call ρ $rho$ . Similarly, we denote also by ρ $rho$ the induced metric on the universal cover Σ ∼ $tilde{Sigma }$ . Definition.A ρ $rho$ -geodesic α : R → Σ ̂ $alpha :mathbb {R}rightarrow hat{Sigma }$ whose image is not contained in ∂ Σ ̂ $partial hat{Sigma }$ is as simple as possible if Before going any further, we note that non-trivial closed ρ hyp $rho _{operatorname{hyp}}$ -geodesics γ : S 1 → O $gamma :mathbb {S}^1rightarrow operatorname{O}$ have representatives η : S 1 → Σ ⊂ O $eta :mathbb {S}^1rightarrow Sigma subset operatorname{O}$ that are as simple as possible. It suffices to choose η ⊂ Σ $eta subset Sigma$ to be a shortest representative of γ $gamma$ . Indeed, the fact that η $eta$ is shortest implies that its lifts to Σ ̂ $hat{Sigma }$ have no bigons, showing that η $eta$ is as simple as possible. We record this fact for later use: Lemma 3.1.Every ρ hyp $rho _{operatorname{hyp}}$ -geodesic γ : S 1 → O $gamma :mathbb {S}^1rightarrow operatorname{O}$ is freely homotopic, in the category of orbifolds, to a ρ $rho$ -geodesic η : S 1 → Σ ⊂ O $eta :mathbb {S}^1rightarrow Sigma subset operatorname{O}$ which is as simple as possible. The reader might be wondering why instead of simply speaking of shortest representatives we choose something as clumsy as “as simple as possible.” The reason is that the latter property is mapping class group invariant: Lemma 3.2.If η ⊂ Σ $eta subset Sigma$ is as simple as possible, then ϕ ( η ) $phi (eta )$ is also as simple as possible for every ϕ ∈ PMap ( Σ ) $phi in operatorname{PMap}(Sigma )$ . Proof.Abusing notation, denote the ρ $rho$ -geodesic freely homotopic to η $eta$ by the same letter. To determine ϕ ( η ) $phi (eta )$ choose first a representative φ ∈ Diff ( Σ ) $varphi in operatorname{Diff}(Sigma )$ of the mapping class ϕ $phi$ and let φ ∗ : π 1 ( Σ ) → π 1 ( Σ ) $varphi _*:pi _1(Sigma )rightarrow pi _1(Sigma )$ be the homomorphism induced by φ $varphi$ — we can always choose φ $varphi$ so that it fixes some point and take that point as the base point for the fundamental group. Note that φ ∗ $varphi _*$ preserves the normal subgroup of π 1 ( Σ ) $pi _1(Sigma )$ generated by loops freely homotopic into ∂ Σ ∖ ∂ O $partial Sigma setminus partial operatorname{O}$ and that π 1 ( Σ ̂ ) ⊂ π 1 ( Σ ) $pi _1(hat{Sigma })subset pi _1(Sigma )$ is nothing other than this subgroup. We get that φ $varphi$ lifts to Σ ̂ $hat{Sigma }$ , or more precisely, that there is a φ ∗ $varphi _*$ -equivariant lift φ ̂ ∈ Diff ( Σ ̂ ) $hat{varphi }in operatorname{Diff}(hat{Sigma })$ . Now, if η ̂ $hat{eta }$ is a lift of η $eta$ to Σ ̂ $hat{Sigma }$ , then we have for all g ∈ Γ $gin Gamma$ that It follows that the image of φ ( η ) $varphi (eta )$ in Σ $Sigma$ has no bigons. The same is true for ( ϕ ( η ) ) ∗ $(phi (eta ))_*$ , the geodesic in ( Σ , ρ ) $(Sigma , rho )$ freely homotopic to φ ( η ) $varphi (eta )$ . Now, [10, Theorem 2.1] implies that these two curves are not only freely homotopic to each other but also transversely freely homotopic to each other. This means in particular that intersection points are neither destroyed nor created during the homotopy. Hence, each lift of ( ϕ ( η ) ) ∗ $(phi (eta ))_*$ to Σ ̂ $hat{Sigma }$ meets its individual Γ $Gamma$ -translates in at most one point. In other words, ϕ ( η ) $phi (eta )$ is as simple as possible. □ $Box$ We are now ready to state the key technical result of this paper. Proposition 3.3.Let O $operatorname{O}$ be as in the statement of Theorem 1.2, Σ ̂ $hat{Sigma }$ as in (2.1), and ρ $rho$ the metric on Σ ̂ $hat{Sigma }$ constructed in Section 2.7. There exists A ⩾ 1 $Ageqslant 1$ such that any unit speed ρ $rho$ -geodesic α : R → Σ ̂ $alpha : mathbb {R}rightarrow hat{Sigma }$ which is (1) a quasigeodesic in ( O ∼ , ρ hyp ) $(tilde{operatorname{O}},rho _{operatorname{hyp}})$ and (2) as simple as possible is actually A $A$ -quasigeodesic in ( O ∼ , ρ hyp ) $(tilde{operatorname{O}},rho _{operatorname{hyp}})$ . In this section we prove Theorem 1.2 assuming Proposition 3.3. However, before doing so we have to recall a few facts about currents and about Mirzakhani's counting theorem. Remark.We insist that X / G $X/G$ , and thus our orbifold O $operatorname{O}$ , is compact because this is what guarantees that the space C or ( X / G ) $mathcal {C}^{operatorname{or}}(X/G)$ is locally compact. There are plenty of currents. In fact there is a natural homeomorphism between C or ( X / G ) $mathcal {C}^{operatorname{or}}(X/G)$ and the space of geodesic flow invariant Radon measures on the projectivized unit tangent bundle P T 1 X / G $PT^1X/G$ supported by the set of bi-infinite orbits. For example, every primitive closed unit speed geodesic γ $gamma$ in G ( X ) $mathcal {G}(X)$ , or equivalently every unoriented periodic orbit of the geodesic flow, yields a geodesic flow invariant measure on P T 1 X / G $PT^1X/G$ : the measure of U $U$ is the arc length of γ ∩ U $gamma cap U$ . The current associated to this measure is called the counting current associated to the geodesic γ $gamma$ . The counting current determines the original geodesic γ $gamma$ — this justifies referring to the current and the geodesic by the same letter — and the name is explained because, for a set of geodesics V ⊂ G ( X ) $Vsubset mathcal {G}(X)$ the value of γ ( V ) $gamma (V)$ is nothing other than the number of lifts of γ $gamma$ to X $X$ which belong to V $V$ . Note that every essential curve γ $gamma$ in X / G $X/G$ (in the sense that we gave to the word essential at the end of Section 2.3) is freely homotopic to a unique geodesic γ ∗ $gamma _*$ in X / G $X/G$ . In this case we denote the associated counting current by γ $gamma$ instead of γ ∗ $gamma _*$ . We hope that this will not cause any confusion. Remark.If the action of G $G$ on X $X$ is free, then we drop the superscript “ or $operatorname{or}$ .” For example, we write C ( Σ ) $mathcal {C}(Sigma )$ instead of C or ( Σ ) $mathcal {C}^{operatorname{or}}(Sigma )$ . We use this superscript to avoid mixing up currents for the orbifold X / G $X/G$ and currents for the underlying topological surface. Currents were introduced by Bonahon and we refer to his papers [2-4] for details and background. See also [1]. However, although all these sources are highly recommended, the reader will not be surprised on hearing that we will mostly follow the same notation and terminology as in our book [7]. As we mentioned already in the introduction, Theorem 1.2 is well known in the case that we are working with surfaces instead of orbifolds. In that case we have the following result [7, Theorem 8.1]. Mirzakhani's counting theorem.Let Σ $Sigma$ be a compact connected orientable surface of genus g $g$ and with r $r$ boundary components and suppose that 3 g − 3 + r > 0 $3g-3+r>0$ . Let also η 0 ⊂ Σ $eta _0subset Sigma$ be a homotopically primitive essential curve. Then there are constants c PMap ( η 0 ) , b g , r PMap > 0 $mathfrak {c}^{operatorname{PMap}}(eta _0),mathfrak {b}^{operatorname{PMap}}_{g,r}>0$ such that Remark.The counting theorem remains true if we replace the pure mapping class group by any other finite index subgroup of the mapping class group. However, the obtained multiple of the Thurston measure depends on the subgroup in question. This explains the superscript PMap $operatorname{PMap}$ in the constants c PMap ( η 0 ) $mathfrak {c}^{operatorname{PMap}}(eta _0)$ and b g , r PMap $mathfrak {b}^{operatorname{PMap}}_{g,r}$ . See [7, Exercise 8.2] for explicit formulas for the dependence of the constants on the chosen subgroup of the mapping class group. We prove now our main theorem assuming Proposition 3.3. As mentioned earlier, the proposition will be proved in Section 6. Theorem 1.2.Let O $operatorname{O}$ be a compact orientable non-exceptional hyperbolic orbifold with possibly empty totally geodesic boundary and let C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ be the associated space of geodesic currents. There is a Radon measure m Thu $mathfrak {m}_{operatorname{Thu}}$ on C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ such that for any γ 0 ∈ S or ( O ) $gamma _0in mathcal {S}^{operatorname{or}}(operatorname{O})$ we have Lemma 4.1.If η 0 : S 1 → ( Σ , ρ ) $eta _0:mathbb {S}^1rightarrow (Sigma ,rho )$ is any essential ρ $rho$ -geodesic which when considered as a map into O $operatorname{O}$ is as simple as possible, then the measure Rather, we have proved Theorem 1.2 while assuming Proposition 3.3. Anyways, before proving the proposition let us prove the other theorems mentioned in the introduction and comment briefly on the measure m Thu $mathfrak {m}_{operatorname{Thu}}$ . Remark.It also seems probable that one can recover the measure m Thu S , H $mathfrak {m}_{operatorname{Thu}}^{S,H}$ as a multiple of the measure on M L ( S ) H $mathcal {M}mathcal {L}(S)^H$ obtained by taking a suitable power of the restriction to that subspace of the Thurston symplectic form on M L ( S ) $mathcal {M}mathcal {L}(S)$ . It would be interesting to do as in [16] and figure out the precise multiple. Anyways, the reader having just the present paper in mind can ignore these past comments and just continue thinking of m Thu $mathfrak {m}_{operatorname{Thu}}$ as given in the proof of Theorem 1.2. Now we prove Theorem 1.1 and Theorem 1.3 from the introduction. Let us start with the latter. Theorem 1.3.Let O $operatorname{O}$ be a compact orientable hyperbolic orbifold with possibly empty totally geodesic boundary and let C or ( O ) $mathcal {C}^{operatorname{or}}(operatorname{O})$ be the associated space of geodesic currents. Then the limit Proof.Noting that there is nothing to prove if O $operatorname{O}$ is exceptional, suppose that this is not the case. For any such homogenous function F : C or ( O ) → R $F:mathcal {C}^{operatorname{or}}(operatorname{O})rightarrow mathbb {R}$ we have We come now to Theorem 1.1. Theorem 1.1.Let Γ ⊂ PSL 2 R $Gamma subset operatorname{PSL}_2mathbb {R}$ be a non-elementary finitely generated discrete subgroup and O = H 2 / Γ $operatorname{O}=mathbb {H}^2/Gamma$ the associated 2-dimensional hyperbolic orbifold. Then the limit Proof.We might once again assume that O $operatorname{O}$ is not exceptional. Let then O ¯ $bar{operatorname{O}}$ be a compact hyperbolic orbifold with interior homeomorphic to O $operatorname{O}$ , consider γ 0 ∈ S or ( O ) $gamma _0in mathcal {S}^{operatorname{or}}(operatorname{O})$ as an element in S or ( O ¯ ) $mathcal {S}^{operatorname{or}}(bar{operatorname{O}})$ , and apply Theorem 1.2 to get Lemma 5.1.If γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ is a ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment, then we have We now prove the lemma. Proof.Set X 0 = N hyp ( γ , r ) $X_0=mathcal {N}_{operatorname{hyp}}(gamma ,r)$ and for p ∈ P ( γ , r ) $pin mathcal {P}(gamma ,r)$ let Y p $Y_p$ be the ρ hyp $rho _{operatorname{hyp}}$ -convex hull of the union of N hyp ( γ , r ) ∩ Σ ̂ $mathcal {N}_{operatorname{hyp}}(gamma ,r)cap hat{Sigma }$ and B hyp ( p , 2 δ ) $B_{operatorname{hyp}}(p,2delta )$ . Since r ⩾ 50 ε $rgeqslant 50epsilon$ then we get from (C4) that X p = Y p ∖ X 0 ⊂ B hyp ( p , ε 2 ) $X_p=Y_psetminus X_0subset B_{operatorname{hyp}}(p,frac{epsilon }{2})$ . This implies that the collection of sets { X p with p ∈ P ( γ , r ) } $lbrace X_ptext{ with }pin mathcal {P}(gamma ,r)rbrace$ is locally finite and that X p ∩ X q = ∅ $X_pcap X_q=emptyset$ for all distinct p , q ∈ P ( γ , r ) $p,qin mathcal {P}(gamma , r)$ . We get thus from (*) that Continuing with the same notation, let γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ be a ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment and let p ∈ O ∼ ∖ γ $pin tilde{operatorname{O}}setminus gamma$ be a point not on γ $gamma$ . Under the γ $gamma$ -outgoing ray at p $p$ we understand the ρ hyp $rho _{operatorname{hyp}}$ -geodesic ray starting at p $p$ in the direction of the gradient of the function d hyp ( γ , · ) $d_{operatorname{hyp}}(gamma ,cdot )$ — that is, the ray that p $p$ would follow to escape from γ $gamma$ at the fastest possible rate. The following lemma asserts that the γ $gamma$ -height of η $eta$ agrees, up to a small error, with the maximal d hyp $d_{operatorname{hyp}}$ -distance to γ $gamma$ from points in η $eta$ . Lemma 5.2.Let γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ and η ⊂ Σ ̂ $eta subset hat{Sigma }$ be a ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment and a ρ $rho$ -geodesics segment, both with the same endpoints. Then we have Proof.Let Q $mathcal {Q}$ be the set of those singular points p ∈ sing ( Γ ) $pin operatorname{sing}(Gamma )$ whose γ $gamma$ -outgoing ray meets η $eta$ . Set r 0 = max { 50 ε , h γ ( η ) } $r_0=max lbrace 50epsilon ,h_gamma (eta )rbrace$ and note that Q ⊂ P ( γ , r 0 ) $mathcal {Q}subset mathcal {P}(gamma ,r_0)$ . The geodesic η $eta$ is homotopic in Σ ̂ $hat{Sigma }$ and while fixing its endpoints to a curve contained in Our next goal is to establish the following fact. Lemma 5.3.Let γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ and η ⊂ Σ ̂ $eta subset hat{Sigma }$ be, respectively, a ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment and a simple ρ $rho$ -geodesic segment, such that both segments have the same endpoints ∂ γ = ∂ η $partial gamma =partial eta$ . Suppose that at least one of the following holds: Remark.It follows by a limiting argument and Lemma 5.3 that, if we replace the “ max $max$ ” in the definition of h γ ( η ) $h_{gamma }(eta )$ by a “ sup $sup$ ,” then the lemma also holds when γ $gamma$ and η $eta$ are a complete ρ hyp $rho _{operatorname{hyp}}$ -geodesic and a complete simple ρ $rho$ -geodesic which have the same endpoints in ∂ ∞ O ∼ $partial _{infty }{tilde{operatorname{O}}}$ , the boundary at infinity of O ∼ $tilde{operatorname{O}}$ . To see that this is the case parametrize η : R → Σ ̂ $eta :mathbb {R}rightarrow hat{Sigma }$ , let γ n $gamma _n$ be the ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment with endpoints η ( − n ) $eta (-n)$ and η ( n ) $eta (n)$ , set η n = η ( [ − n , n ] ) $eta _n=eta ([-n,n])$ , and note that Proof.Starting with the proof of Lemma 5.3, suppose that γ $gamma$ and η $eta$ satisfy one of the two possible conditions in the statement. As a first observation note that if γ $gamma$ and η $eta$ satisfy (a) (respectively, (b)) and meet in a point other than in their end points, then there are subsegments γ ′ ⊂ γ $gamma ^{prime }subset gamma$ and η ′ ⊂ η $eta ^{prime }subset eta$ with ∂ γ ′ = ∂ η ′ $partial gamma ^{prime }=partial eta ^{prime }$ , which still satisfy (a) (respectively, (b)) and such that γ ′ $gamma ^{prime }$ and η ′ $eta ^{prime }$ meet only at their endpoints. This means that we can assume without loss of generality that the loop obtained by concatenating γ $gamma$ and η $eta$ is simple. Or said differently that γ $gamma$ and η $eta$ bound a disk Δ $Delta$ in H 2 $mathbb {H}^2$ . Claim 1.There are a ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment γ ¯ $bar{gamma }$ and a subsegment η ¯ ⊂ η $bar{eta }subset eta$ satisfying the following properties. We suggest that at first, instead of studying the proof of the claim, the reader spends some time looking at Figure 5. Proof of Claim 1.If the disk Δ $Delta$ bounded by the concatenation of γ $gamma$ and η $eta$ satisfies (3), then we have nothing to prove. If this is not the case, then we will find a hyperbolic geodesic segment γ ′ $gamma ^{prime }$ and a subsegment η ′ ⊂ η $eta ^{prime }subset eta$ satisfying (1) and (2), and such that η ′ $eta ^{prime }$ is at least ε $epsilon$ -shorter than η $eta$ , that is, ℓ ρ ( η ′ ) ⩽ ℓ ρ ( η ) − ε $ell _rho (eta ^{prime })leqslant ell _rho (eta )-epsilon$ . Now, if the disk Δ ′ $Delta ^{prime }$ associated to γ ′ $gamma ^{prime }$ and η ′ $eta ^{prime }$ satisfies (3), then we are done. Otherwise we iterate our procedure. But this process can only be repeated finitely many times because at each step we lose a definite amount of length, and the length of the original segment η $eta$ is finite. Let us see how we find γ ′ $gamma ^{prime }$ and η ′ $eta ^{prime }$ . We start by taking a point p ∈ sing ( Γ ) $pin operatorname{sing}(Gamma )$ such that the γ $gamma$ -outgoing ray σ $sigma$ at p $p$ meets η $eta$ and with d hyp ( p , γ ) = h γ ( η ) $d_{operatorname{hyp}}(p,gamma )=h_gamma (eta )$ . Note that Lemma 5.2 implies that all intersections of σ $sigma$ and η $eta$ happen in the annulus B hyp ( p , ε ) ∩ Σ ̂ = B hyp ( p , ε ) ∖ B hyp ( p , δ ) $B_{operatorname{hyp}}(p,epsilon )cap hat{Sigma }=B_{operatorname{hyp}}(p,epsilon )setminus B_{operatorname{hyp}}(p,delta )$ . Since the outgoing ray σ $sigma$ intersects η $eta$ and we are assuming that p ∉ Δ $pnotin Delta$ , σ $sigma$ must intersect η $eta$ at least twice. We deduce that there is a closed subsegment By construction the pair γ ′ , η ′ $gamma ^{prime },eta ^{prime }$ satisfies (1). Moreover, since η $eta$ has to travel at least distance 49 ε $49epsilon$ to go from γ $gamma$ to B hyp ( p , ε ) $B_{operatorname{hyp}}(p,epsilon )$ and since the metrics ρ $rho$ and ρ hyp $rho _{operatorname{hyp}}$ agree on B hyp ( p , 50 ε ) ∖ B hyp ( p , δ ) $B_{operatorname{hyp}}(p,50epsilon )setminus B_{operatorname{hyp}}(p,delta )$ , we get that Continuing with the proof of Lemma 5.3 and with notation as in Claim 1, choose g ∈ Stab Γ ( p ) $gin operatorname{Stab}_{Gamma }(p)$ with rotation angle θ ∈ [ 2 π 3 , 4 π 3 ] $theta in [frac{2pi }{3}, frac{4pi }{3}]$ . Note that such g $g$ always exists. Claim 2.We have g ± 1 ( γ ¯ ) ∩ Δ ¯ = ∅ $g^{pm 1}(bar{gamma })cap bar{Delta }=emptyset$ . Proof of Claim 2.Suppose first that the pair ( γ ¯ , η ¯ ) $(bar{gamma },bar{eta })$ satisfies (a), meaning that We are now finally ready to prove the remaining proposition. Proposition 3.3.Let O $operatorname{O}$ be as in the statement of Theorem 1.2, Σ ̂ $hat{Sigma }$ as in (2.1), and ρ $rho$ the metric on Σ ̂ $hat{Sigma }$ constructed in Section 2.7. There exists A ⩾ 1 $Ageqslant 1$ such that any unit speed ρ $rho$ -geodesic α : R → Σ ̂ $alpha : mathbb {R}rightarrow hat{Sigma }$ which is (1) a quasigeodesic in ( O ∼ , ρ hyp ) $(tilde{operatorname{O}},rho _{operatorname{hyp}})$ and (2) as simple as possible is actually A $A$ -quasigeodesic in ( O ∼ , ρ hyp ) $(tilde{operatorname{O}},rho _{operatorname{hyp}})$ . Proof.Note first that compactness of Σ ̂ / Γ $hat{Sigma }/Gamma$ , together with Γ $Gamma$ -invariance of the metric ρ $rho$ , implies that the inclusion ( Σ ̂ , ρ ) ↪ ( O ∼ , ρ hyp ) $(hat{Sigma },rho )hookrightarrow (tilde{operatorname{O}},rho _{operatorname{hyp}})$ is locally A 0 $A_0$ -bi-Lipschitz for some A 0 $A_0$ . It follows in particular that for all s , t ∈ R $s,tin mathbb {R}$ we have As a first step we choose a Γ $Gamma$ -invariant triangulation T $mathcal {T}$ of O ∼ $tilde{operatorname{O}}$ whose edges T $mathcal {T}$ are ρ hyp $rho _{operatorname{hyp}}$ -geodesic segments of length at most ε $epsilon$ . Consider the set We get from (b) that a geodesic α $alpha$ as in the statement never spends much time without meeting one of the edges in E $mathcal {E}$ . We prove next that once such an α : R → Σ ̂ $alpha : mathbb {R}rightarrow hat{Sigma }$ leaves e ∈ E $ein mathcal {E}$ , it never comes back to e $e$ . Indeed, suppose that s < t $s<t$ are such that α ( s ) , α ( t ) ∈ e $alpha (s),alpha (t)in e$ for some e ∈ E $ein mathcal {E}$ . Denote by γ $gamma$ the subsegment of e $e$ between α ( s ) $alpha (s)$ and α ( t ) $alpha (t)$ and let η = α [ s , t ] $eta =alpha [s,t]$ . We claim first that η ⊂ N hyp ( γ , 55 ε ) $eta subset mathcal {N}_{operatorname{hyp}}(gamma ,55epsilon )$ . Otherwise we get from Lemma 5.2 that h γ ( η ) > 50 ε $h_gamma (eta )>50epsilon$ and then from Lemma 5.3 that there is g ∈ Γ $gin Gamma$ such that | η ∩ g η | ⩾ 2 $vert eta cap geta vert geqslant 2$ , contradicting the assumption that α $alpha$ is as simple as possible. We have thus proved that η ⊂ N hyp ( γ , 55 ε ) $eta subset mathcal {N}_{operatorname{hyp}}(gamma ,55epsilon )$ . But noting that N hyp ( γ , 55 ε ) ⊂ N hyp ( e , 55 ε ) ⊂ Σ ̂ $mathcal {N}_{operatorname{hyp}}(gamma ,55epsilon )subset mathcal {N}_{operatorname{hyp}}(e,55epsilon )subset hat{Sigma }$ is contractible and that both metrics ρ $rho$ and ρ hyp $rho _{operatorname{hyp}}$ agree thereon, we deduce that γ = η $gamma =eta$ because both are geodesic segments with the same endpoints. We have established the following key fact. The reader surely can at this point imagine how (b) and (c) interplay, but might still be wondering why we bothered to state (a) at all. Well, the reason is coming. Still assuming that α : R → Σ ̂ $alpha :mathbb {R}rightarrow hat{Sigma }$ is a ρ $rho$ -geodesic as in the statement, suppose that s , t ∈ R $s,tin mathbb {R}$ are such that d hyp ( α ( s ) , α ( t ) ) ⩽ 4 $d_{operatorname{hyp}}(alpha (s),alpha (t))leqslant 4$ , let γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ be the ρ hyp $rho _{operatorname{hyp}}$ -geodesic segment joining α ( s ) $alpha (s)$ and α ( t ) $alpha (t)$ , and finally let p $p$ be the midpoint of γ $gamma$ . Since α $alpha$ is as simple as possible, we get from Lemma 5.3 that h γ ( α [ s , t ] ) ⩽ 1 $h_gamma (alpha [s,t])leqslant 1$ . Lemma 5.2 yields then that We are almost at the end of the proof of the proposition. Recalling that α $alpha$ as in the statement of the proposition is a quasigeodesic, let γ ⊂ O ∼ $gamma subset tilde{operatorname{O}}$ be the hyperbolic geodesic with the same endpoints as α $alpha$ , and let The claim follows then from (6.1) and (6.2) with A = max { A 0 , A 1 , 1 } $A=max lbrace A_0,A_1, 1rbrace$ . □ $Box$ This has been one of those projects that for whatever reason take a long time to be completed. So long in fact that it is be impossible to make a comprehensive list of everyone we owe our gratitude to, and wishing not to be unfair we thank nobody — ingen nämnd ingen glömd. With one exception, because the first author has not forgotten that during the start of the project she was supported by Pekka Pankka's Academy of Finland project #297258 at the University of Helsinki. The first author gratefully acknowledges support from EPSRC grant EP/T015926/1. The Transactions of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics." @default.
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