On type-preserving representations of the four-punctured sphere group
OOn type-preserving representations of the four-puncturedsphere group
Tian Yang
Abstract
We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation ρ : π (Σ g,n ) → P SL (2; R ) of a punctured surface group sends every non-peripheral simple closed curveto a hyperbolic element, then must ρ be Fuchsian. The counterexamples come from relative Euler class ± representations of the four-punctured sphere group. We also show that the mapping class group action on eachnon-extremal component of the character space of type-preserving representations of the four-punctured spheregroup is ergodic, which confirms a conjecture of Goldman for this case. The main tool we use is Kashaev-Penner’slengths coordinates of the decorated character spaces. Let Σ g be an oriented closed surface of genus g (cid:62) . The
P SL (2 , R ) -representation space R (Σ g ) is the space ofgroup homomorphisms ρ : π (Σ g ) → P SL (2 , R ) from the fundamental group of Σ g into P SL (2 , R ) , endowedwith the compact open topology. The Euler class e ( ρ ) of ρ is the Euler class of the associated S -bundle on Σ g , which satisfies the Milnor-Wood inequality − g (cid:54) e ( ρ ) (cid:54) g − . In [9], Goldman proved that equality holds ifand only if ρ is Fuchsian, i.e., discrete and faithful; and in [11], he proved that the connected components of R (Σ g ) are indexed by the Euler classes. I.e., for each integer k with | k | (cid:54) g − , the representations of Euler class k exist and form a connected component of R (Σ g ) . The Lie group
P SL (2 , R ) acts on R (Σ g ) by conjugation, andthe quotient space M (Σ g ) = R (Σ g ) /P SL (2 , R ) is the character space of Σ g . Since the Euler classes are preserved by conjugation, the connected components of M (Σ g ) are also indexed by the Euler classes, i.e., M (Σ g ) = g − (cid:97) k =2 − g M k (Σ g ) , where M k (Σ g ) is the space of conjugacy classes of representations of Euler class k . By the results of Goldman [9,11], the extremal components M ± (2 − g ) (Σ g ) are respectively identified with the Teichm¨uller space of Σ g and thatof Σ g endowed with the opposite orientation.The mapping class group M od (Σ g ) of Σ g is the group of isotopy classes of orientation preserving self-diffeomorphisms of Σ g . By the Dehn-Nielsen Theorem,
M od (Σ g ) is naturally isomorphic to the group of positiveouter-automorphisms Out + ( π (Σ g )) , which acts on M (Σ g ) preserving the Euler classes. Therefore, M od (Σ g ) acts on each connected component of M (Σ g ) . It is well known (see e.g. Fricke [8]) that the
M od (Σ g ) -actionis properly discontinuous on the extremal components M ± (2 − g ) (Σ g ) , i.e., the Teichm¨uller spaces, and the quo-tients are the Riemann moduli spaces of complex structures on Σ g . On the non-extremal components M k (Σ g ) , | k | < g − , Goldman conjectured in [14] that the
M od (Σ g ) -action is ergodic with respect to the measure inducedby the Goldman symplectic form [10].Closely related to Goldman’s conjecture is a question of Bowditch [4], Question C, that whether for eachnon-elementary and non-extremal (i.e. non-Fuchsian) representation ρ in R (Σ g ) , there exists a simple closedcurve γ on Σ g such that ρ ([ γ ]) is an elliptic or a parabolic element of P SL (2 , R ) . Recall that a representationis non-elementary if its image is Zariski-dense in
P SL (2 , R ) . Recently, March´e-Wolff [23] show that an affirma-tive answer to Bowditch’s question implies that Goldman’s conjecture is true. More precisely, they show that for1 a r X i v : . [ m a t h . G T ] J u l g, k ) (cid:54) = (2 , , M od (Σ g ) acts ergodically on the subset of M k (Σ g ) consisting of representations that send somesimple closed curve on Σ g to an elliptic or parabolic element. Therefore, when ( g, k ) (cid:54) = (2 , , the M od (Σ g ) -action on M k (Σ g ) is ergodic if and only if the subset above has full measure in M k (Σ g ) . In the same work, theyanswer Bowditch’s question affirmatively for the genus surface Σ , implying the ergodicity of the M od (Σ ) -action on M ± (Σ ) . For the action on the component M (Σ ) , they find two M od (Σ ) -invariant open subsetsdue to the existence of the hyper-elliptic involution, and show that on each of them the M od (Σ ) -action is ergodic.For higher genus surfaces Σ g , g (cid:62) , Souto [27] recently gives an affirmative answer to Bowditch’s question forthe Euler class representations, proving the ergodicity of the M od (Σ g ) -action on M (Σ g ) . For g (cid:62) and k (cid:54) = 0 , both Bowditch’s question and Goldman’s conjecture are still open.Bowditch’s question was originally asked for the type-preserving representations of punctured surface groups.Recall that a punctured surface Σ g,n of genus g with n punctures is a closed surface Σ g with n points removed.Through out this paper, we required that the Euler characteristic of Σ g,n is negative. A peripheral element of π (Σ g,n ) is an element that is represented by a curve freely homotopic to a circle that goes around a single punctureof Σ g,n . A representation ρ : π (Σ g,n ) → P SL (2 , R ) is called type-preserving if it sends every peripheral elementof π (Σ g,n ) to a parabolic element of P SL (2 , R ) . In [4], Question C asks whether it is true that if a non-elementarytype-preserving representation of a punctured surface group sends every non-peripheral simple closed curve to ahyperbolic element, then ρ must be Fuchsian.The main result of this paper gives counterexamples to this question. To state the result, we recall that thereis the notion of relative Euler class e ( ρ ) of a type-preserving representation ρ that satisfies the Milnor-Woodinequality | e ( ρ ) | (cid:54) g − n, and equality holds if and only if ρ is Fuchsian (see [9, 11] and also Proposition A.1). Theorem 1.1.
There are uncountably many non-elementary type-preserving representations ρ : π (Σ , ) → P SL (2 , R ) with relative Euler class e ( ρ ) = ± that send every non-peripheral simple closed curve to a hyperbolicelement. In particular, these representations are not Fuchsian. Our method is to use Penner’s lengths coordinates for the decorated character space defined by Kashaev in[18]. Briefly speaking, decorated character space of a punctured surface is the space of conjugacy classes of dec-orated representations , namely, non-elementary type-preserving representations together with an assignment of horocycles to the punctures. The lengths coordinate of a decorated representation depend on a choice of an idealtriangulation of the surface, and consists of the λ -lengths of the edges determined by the horocycles, and of the signs of the ideal triangles determined by the representation. The decorated Teichm¨uller space is a connected com-ponent of the decorated character space, and the restriction of the lengths coordinates to this component coincidewith Penner’s lengths coordinates. (See [18, 19] or Section 2 for more details.) A key ingredient in the proof isFormula (3.1) of the traces of closed curves in the lengths coordinates, found in [28] by Sun and the author. Witha careful choice of an ideal triangulation of the four-punctured sphere, called a tetrahedral triangulation , we showthat the traces of three distinguished simple closed curves are greater than in the absolute value if and only if the λ -lengths of edges in this triangulation satisfy certain anti-triangular inequalities. We then show that each simpleclosed curve is distinguished in some tetrahedral triangulation, and all tetrahedral triangulations are related by a se-quence of moves, called the simultaneous diagonal switches . By the change of λ -lengths formula (Proposition 2.3),we show that the anti-triangular inequalities are preserved by the simultaneous diagonal switches. Therefore, if thethree distinguished simple closed curves are hyperbolic, then all the simple closed curves are hyperbolic. Finally,we show that there are uncountably many choices of the λ -lengths that satisfy the anti-triangular inequalities.A consequence of Formula (3.1) is Theorem 3.5 that each non-Fuchsian type-preserving representation is dom-inated by a Fuchsian one, in the sense that the traces of the simple closed curves of the former representation areless than or equal to those of the later in the absolute value. This is a counterpart of the result of Gueritaud-Kassel-Wolff [16] and Deroin-Tholozan [6], where they consider dominance of closed surface group representations.Using the same technique, we also give an affirmative answer to Bowditch’s question for the relative Eulerclass type-preserving representations of the four-punctured sphere group. Theorem 1.2.
Every non-elementary type-preserving representation ρ : π (Σ , ) → P SL (2 , R ) with relativeEuler class e ( ρ ) = 0 sends some non-peripheral simple closed curve to an elliptic or parabolic element.
2n contrast with the connected components of the character space of a closed surface, those of a puncturedsurface are more subtle to describe. For Σ g,n with n (cid:54) = 0 , denote by M k (Σ g,n ) be the space of conjugacy classesof type-preserving representations with relative Euler class k. As explained in [18], M k (Σ g,n ) can be either emptyor non-connected. For example, M (Σ , ) = M (Σ , ) = ∅ . The non-connectedness of M k (Σ g,n ) comes fromthe existence of two distinct conjugacy classes of parabolic elements of P SL (2 , R ) . More precisely, each parabolicelement of
P SL (2 , R ) is up to ± I conjugate to an upper triangular matrix with trace , and its conjugacy class isdistinguished by whether the sign of the non-zero off diagonal element is positive or negative. Therefore, two type-preserving representations of the same relative Euler class which respectively send the same peripheral element intodifferent conjugacy classes of parabolic elements cannot be in the same connected components. Throughout thispaper, we respectively call the two conjugacy class of parabolic elements the positive and the negative conjugacyclasses. For a type-preserving representation ρ : π (Σ g,n ) → P SL (2 , R ) , we say that the sign of a puncture v is positive , denoted by s ( v ) = 1 , if ρ sends a peripheral element around this puncture into the positive conjugacyclass of parabolic elements, and is negative , denoted by s ( v ) = − , if otherwise. For an s ∈ {± } n , we denote by M sk (Σ g,n ) the space of conjugacy classes of type-preserving representations with relative Euler class k and signsof the punctures s. It is conjectured in [18] that each M sk (Σ g,n ) , if non-empty, is connected. The result confirmsthis for the four-punctured sphere. Theorem 1.3.
Let s ∈ {± } . Then(1) M s (Σ , ) is non-empty if and only if s contains exactly two − and two , (2) M s (Σ , ) is non-empty if and only if s contains at most one − , (3) M s − (Σ , ) is non-empty if and only if s contains at most one , and(4) all the non-empty spaces above are connected. As a consequence of Theorem 1.3, M (Σ , ) has six connected components and each of M ± (Σ , ) has fiveconnected components. The main tool we use in the proof is still the lengths coordinates; and we hope that thetechnique could be used for the other punctured surfaces.The mapping class group M od (Σ g,n ) of a punctured surface Σ g,n is the group of relative isotopy classesof orientation preserving self-diffeomorphisms of Σ g,n that fix the punctures. By the Dehn-Nielsen Theorem, M od (Σ g,n ) is isomorphic to the group of positive outer-automorphisms Out + ( π (Σ g,n )) that preserve the cyclicsubgroups of π (Σ g,n ) generated by the peripheral elements, and hence acts on M (Σ g,n ) preserving the relativeEuler classes and the signs of the punctures. Therefore, for any integer k with | k | (cid:54) g − n and for any s ∈ {± } n , M od (Σ g,n ) acts on M sk (Σ g,n ) . For the four-punctured sphere, we have
Theorem 1.4.
The
M od (Σ , ) -action on each non-extremal connected component of M (Σ , ) is ergodic. By March´e-Wolff [23], it is not surprising that the
M od (Σ , ) -action is ergodic on the connected componentsof M (Σ , ) where Bowditch’s question has an affirmative answer. A new and unexpected phenomenon Theorem1.4 reveals here is that, for punctured surfaces, the action of the mapping class group can still be ergodic when theanswer to Bowditch’s question is negative. Evidenced by Theorem 1.4, we make the following Conjecture 1.5.
The
M od (Σ g,n ) -action is ergodic on each non-extremal connected component of M (Σ g,n ) . The paper is organized as follows. In Section 2, we recall Kashaev’s decorated character spaces and the lengthscoordinates, in Section 3, we obtain a formula of the traces of closed curves in the lengths coordinates, and inSection 4, we introduce tetrahedral triangulations, distinguished simple closed curves and simultaneous diagonalswitches. Then we prove Theorems 1.1 and 1.2, Theorem 1.3 and Theorem1.4 respectively in Sections 5, Section6 and Section 7. It is pointed out by the anonymous referee that the results concerning representations of relativeEuler class ± can be deduced more directly from the results of Goldman in [13], where we present his argumentin Appendix B for the interested readers. Acknowledgments:
The author is grateful to the referee for bringing his attention to the relationship of thiswork with a previous work of Goldman, and for several valuable suggestions to improve the writing of this paper.The author also would like to thank Steven Kerckhoff, Feng Luo and Maryam Mirzakhani for helpful discussionsand suggestions, Ser Peow Tan, Maxime Wolff, Sara Moloni, Fr´ed´eric Palesi and Zhe Sun for discussion and3howing interest, Julien Roger for bringing his attention to Kashaev’s work on the decorated character variety,Ronggang Shi for answering his questions on ergodic theory and Yang Zhou for writing a Python program fortesting some of the author’s ideas.The author is supported by NSF grant DMS-1405066.
We recall the decorated character spaces and the lengths coordinates in this section. The readers are recommendedto read Kashaev’s papers [18, 19] for the original approach and for more details.Let Σ g,n be a punctured surface of genus g with n punctures, and let ρ : π (Σ g,n ) → P SL (2 , R ) be a non-elementary type-preserving representation. A pseudo-developing map D ρ of ρ is a piecewise smooth ρ -equivariantmap from the universal cover of Σ g,n to the hyperbolic plane H . By [9], ρ is the holonomy representation of D ρ . Let ω be the hyperbolic area form of H . Since D ρ is ρ -equivariant, the pull-back -form ( D ρ ) ∗ ω descends to Σ g,n . The relative Euler class e ( ρ ) of ρ could be calculated by e ( ρ ) = 12 π (cid:90) Σ g,n ( D ρ ) ∗ ω. An ideal arc α on Σ g,n is an arc connecting two (possibly the same) punctures. The image D ρ (˜ α ) of a lift ˜ α of α is an arc in H connecting two (possibly the same) points on ∂ H , each of which is the fixed point of the ρ -imageof certain peripheral element of π (Σ g,n ) . We call α ρ -admissible if the two end points of D ρ (˜ α ) are distinct. It iseasy to see that α being ρ -admissible is independent of the choice of the lift ˜ α and the pseudo-developing map D ρ . An ideal triangulation T of Σ g,n consists of a set of disjoint ideal arcs, called the edges, whose complement is adisjoint union of triangles, called the ideal triangles. We call T ρ -admissible if all the edges of T are ρ -admissible.If ρ (cid:48) is conjugate to ρ, then it is easy to see that T is ρ (cid:48) -admissible if and only if it is ρ -admissible. In [18], Kashaevshows that Theorem 2.1 (Kashaev) . For each ideal triangulation T , the set M T (Σ g,n ) = (cid:8) [ ρ ] ∈ M (Σ g,n ) (cid:12)(cid:12) T is ρ -admissible (cid:9) is open and dense in M (Σ g,n ) , and there exist finitely many ideal triangulations T i , i = 1 , . . . , m, such that M (Σ g,n ) = m (cid:91) i =1 M T i (Σ g,n ) . Let Σ g,n and ρ be as above. A decoration of ρ is an assignment of horocycles centered at the fixed points ofthe ρ -image of the peripheral elements of π (Σ g,n ) , one for each, which is invariant under the ρ ( π (Σ g,n )) -action.In the case that the fixed points of the ρ -image of two peripheral elements coincide, which may happen only when ρ is non-Fuchsian, we do not require the corresponding assigned horocycles to be the same. If d is a decorationof ρ and g is an element of P SL (2 , R ) , then the g -image of the horocycles in d form a decoration g · d of theconjugation gρg − of ρ. We call a pair ( ρ, d ) a decorated representation , and call two decorated representations ( ρ, d ) and ( ρ (cid:48) , d (cid:48) ) equivalent if ρ (cid:48) = gρg − and d (cid:48) = g · d for some g ∈ P SL (2 , R ) . The decorated character space of Σ g,n , denoted by M d (Σ g,n ) , is the space of equivalence classes of decorated representations. In [18], Kashaevshows that the projection π : M d (Σ g,n ) → M (Σ g,n ) defined by π ([( ρ, d )]) = [ ρ ] is a principle R V> -bundle,where V is the set of punctures of Σ g,n , and the pre-image of the extremal components are isomorphic as principle R V> -bundles to Penner’s decorated Teichm¨uller space [24].Fixing a ρ -admissible ideal triangulation T and a pseudo-developing map D ρ , the lengths coordinate of ( ρ, d ) consists of the following two parts: the λ -lengths of the edges and the signs of the ideal triangles. The λ -lengthof an edge e of T is defined as follows. Since e is ρ -admissible, for any lift (cid:101) e of e to the universal cover theimage D ρ ( (cid:101) e ) connects to distinct points u and u on ∂ H . The decoration d assigns two horocycles H and H respectively centered at u and u . Let l ( e ) be the signed hyperbolic distance between the two horocycles, i.e., l ( e ) > if H and H are disjoint and l ( e ) (cid:54) if otherwise. Then the λ -length of e in the decorated representation ( ρ, d ) is defined by λ ( e ) = exp l ( e )2 . t of T is defined as follows. Let v , v and v be the vertices of t so that the orientationon t determined by the cyclic order v (cid:55)→ v (cid:55)→ v (cid:55)→ v coincides with the one induced from the orientationof Σ g,n . Let ˜ t be a lift of t to the universal cover, and let (cid:101) v , (cid:101) v and (cid:101) v be the vertices of ˜ t so that (cid:101) v i is a lift of v i , i = 1 , , . Since T is ρ -admissible, the points D ρ ( (cid:101) v ) , D ρ ( (cid:101) v ) and D ρ ( (cid:101) v ) are distinct on ∂ H , and hencedetermine a hyperbolic ideal triangle ∆ in H with them as the ideal vertices. The sign of t is positive, denotedby (cid:15) ( t ) = 1 , if the orientation on ∆ determined by the cyclic oder D ρ ( (cid:101) v ) (cid:55)→ D ρ ( (cid:101) v ) (cid:55)→ D ρ ( (cid:101) v ) (cid:55)→ D ρ ( (cid:101) v ) coincides with the one induced from the orientation of H . Otherwise, the sign of t is negative, and is denoted by (cid:15) ( t ) = − . From the construction, it is easy to see that the λ -lengths λ ( e ) and the signs (cid:15) ( t ) depend only on theequivalence class of ( ρ, d ) . Let T be the set of ideal triangles of T . Then the integral of the pull-back form ( D ρ ) ∗ ω over Σ g,n equals (cid:80) t ∈ T (cid:15) ( t ) π, and the relative Euler class of ρ can be calculated by e ( ρ ) = 12 (cid:88) t ∈ T (cid:15) ( t ) . (2.1)Let V be the set of punctures of Σ g,n and let E be the set of edges of T . Then there is a principle R V> -bundlestructure on R E> defined as follows. For µ ∈ R V> and λ ∈ R E> , we define µ · λ ∈ R E> by ( µ · λ )( e ) = µ ( v ) λ ( e ) µ ( v ) , where v and v are the punctures connected by the edge e. Theorem 2.2 (Kashaev) . Let π : M d (Σ g,n ) → M (Σ g,n ) be the principle R V> -bundle, and let M d T (Σ g,n ) thepre-image of M T (Σ g,n ) . Then M d T (Σ g,n ) = (cid:97) (cid:15) ∈{± } T R ( T , (cid:15) ) , where each R ( T , (cid:15) ) is isomorphic as principle R V> -bundles to an open subset of R E> . The isomorphism is givenby the λ -lengths, and the image of R ( T , (cid:15) ) is the complement of the zeros of certain rational function coming fromthe image of the peripheral elements not being the identity matrix. On M (Σ g,n ) there is the Goldman symplectic form ω W P which restricts to the Weil-Petersson symplecticform on the Teichm¨uller component [10]. By [18, 19], for each ideal triangulation T , the pull-back of ω W P to M d T (Σ g,n ) is expressed in the λ -lengths by π ∗ ω W P = (cid:88) t ∈ T (cid:16) dλ ( e ) ∧ dλ ( e ) λ ( e ) λ ( e ) + dλ ( e ) ∧ dλ ( e ) λ ( e ) λ ( e ) + dλ ( e ) ∧ dλ ( e ) λ ( e ) λ ( e ) (cid:17) , (2.2)where e , e and e are the edges of the ideal triangle t in the cyclic order induced from the orientation of Σ g,n . This formula is first obtained by Penner [25] for the decorated Teichm¨uller space. From (2.2), it is easy to see thatthe measure on each R ( T , (cid:15) ) induced by π ∗ ω W P is in the measure class of the pull-back of the Lebesgue measureof R E> . A diagonal switch at an edge e of T replaces the edge e by the other diagonal of the quadrilateral formed bythe union of the two ideal triangles adjacent to e. By [17], any ideal triangulation can be obtained from another bydoing a finite sequence of diagonal switches. Let ( ρ, d ) be a decorated representation and let T be a ρ -admissibleideal triangulation of Σ g,n . If T (cid:48) is the ideal triangulation of Σ g,n obtained from T by doing a diagonal switch atan edge e, then the ρ -admissibility of T (cid:48) and the lengths coordinate of ( ρ, d ) in T (cid:48) are determined as follows. Let t and t be the two ideal triangles of T adjacent to e, let e (cid:48) be the new edge of T (cid:48) and let t (cid:48) and t (cid:48) be the twoideal triangles in T (cid:48) adjacent to e (cid:48) . We respectively name the edges of the quadrilateral e , . . . , e in the way that e is adjacent to t and t (cid:48) , e is adjacent to t and t (cid:48) , e is adjacent to t and t (cid:48) and e is adjacent to t and t (cid:48) . Then e is opposite to e , and e is opposite to e in the quadrilateral. Proposition 2.3 (Kashaev) . (1) If the signs (cid:15) ( t ) = (cid:15) ( t ) , then T (cid:48) is ρ -admissible. In this case, (cid:15) ( t (cid:48) ) = (cid:15) ( t (cid:48) ) = (cid:15) ( t ) and λ ( e (cid:48) ) = λ ( e ) λ ( e ) + λ ( e ) λ ( e ) λ ( e ) , and the signs of the common ideal triangles and the λ -lengths of the common edges of T and T (cid:48) do not change.(2) If (cid:15) ( t ) (cid:54) = (cid:15) ( t ) , then T (cid:48) is ρ -admissible if and only if λ ( e ) λ ( e ) (cid:54) = λ ( e ) λ ( e ) . In this case, λ ( e ) λ ( e ) < λ ( e ) λ ( e ) , then (cid:15) ( t (cid:48) ) = (cid:15) ( t ) , (cid:15) ( t (cid:48) ) = (cid:15) ( t ) and λ ( e (cid:48) ) = λ ( e ) λ ( e ) − λ ( e ) λ ( e ) λ ( e ) , (2.2) if λ ( e ) λ ( e ) < λ ( e ) λ ( e ) , then (cid:15) ( t (cid:48) ) = (cid:15) ( t ) , (cid:15) ( t (cid:48) ) = (cid:15) ( t ) and λ ( e (cid:48) ) = λ ( e ) λ ( e ) − λ ( e ) λ ( e ) λ ( e ) , and the signs of the common ideal triangles and the λ -lengths of the common edges of T and T (cid:48) do not change, The rule for the signs in (2.1) and (2.2) is that the signs of the ideal triangles adjacent to the shorter edges donot change. This could be seen as follows. If, for example, (cid:15) ( t ) = − , (cid:15) ( t ) = 1 and λ ( e ) λ ( e ) < λ ( e ) λ ( e ) , then the hyperbolic ideal triangle ∆ determined by t is negatively oriented, the hyperbolic ideal triangle ∆ determined by t is positively oriented, and the geodesic arcs a and a determined by e and e intersect. SeeFigure 1. As a consequence, the hyperbolic ideal triangle ∆ (cid:48) determined by t (cid:48) is negatively oriented and thehyperbolic ideal triangle ∆ (cid:48) determined by t (cid:48) is positively oriented, hence (cid:15) ( t (cid:48) ) = − and (cid:15) ( t (cid:48) ) = 1 . The λ -lengths of e (cid:48) follows from Penner’s Ptolemy relation [24] that λ ( e ) λ ( e ) = λ ( e ) λ ( e ) + λ ( e ) λ ( e (cid:48) ) . The othercases could be verified similarly. 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎𝑎 (cid:1) ℍ ℍ ee ee e e eee (cid:1) et t t (cid:1) t (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) Figure 1By Theorems 2.1 and 2.2, the family of open subsets {R ( T , (cid:15) ) } where T goes over all the ideal triangulationsof Σ g,n together with the λ -lengths functions { λ : R ( T , (cid:15) ) → R E> } form a coordinate system of M d (Σ g,n ) , andthe transition functions are given by Proposition 2.3. Theorem 2.4 (Kashaev) . For each relative Euler class k, let M dk (Σ g,n ) be the pre-image of M k (Σ g,n ) under theprojection π : M d (Σ g,n ) → M (Σ g,n ) . Then M dk (Σ g,n ) = (cid:91) T (cid:97) (cid:15) R ( T , (cid:15) ) , where the union is over all the ideal triangulations T and the disjoint union is over all (cid:15) ∈ {± } T such that (cid:80) t ∈ T (cid:15) ( t ) = 2 k. Moreover, M dk (Σ g,n ) ’s are principle R V> -bundles, and are disjoint for different k. A trace formula for closed curves
Throughout this section, we let T be an ideal triangulation of Σ g,n , and let E and T respectively be the set ofedges and ideal triangles of T . Given the lengths coordinate ( λ, (cid:15) ) ∈ R E> × {± } T , up to conjugation, the type-preserving representation ρ : π (Σ g,n ) → P SL (2 , R ) can be reconstructed up to conjugation as follows. Suppose e is an edge of T , and t and t are the two ideal triangles adjacent to e. Let e and e be the other two edges of t and let e and e be the other two edges of t so that the cyclic orders e (cid:55)→ e (cid:55)→ e (cid:55)→ e and e (cid:55)→ e (cid:55)→ e (cid:55)→ e coincide with the one induced from the orientation of Σ g,n . Define the quantity X ( e ) ∈ R > by X ( e ) = λ ( e ) λ ( e ) λ ( e ) λ ( e ) . Note that if ρ is discrete and faithful, then X ( e ) is the shear parameter of the corresponding hyperbolic structureat e. (See e.g. [1].) It is well known that each immersed closed curve on Σ g,n is homotopic to a normal one thattransversely intersects each ideal triangle in simple arcs that connect different edges of the triangle. Let γ be animmersed oriented closed normal curve on Σ g,n . For each edge e intersecting γ, define S ( e ) = (cid:20) X ( e ) X ( e ) − (cid:21) . For each ideal triangle t intersecting γ, define R ( t ) = (cid:20) (cid:15) ( t )0 1 (cid:21) if γ makes a left turn in t (Figure 2 (a)), and define R ( t ) = (cid:20) (cid:15) ( t ) 1 (cid:21) if γ makes a right turn in t (Figure 2 (b)). (a) (b) e e ee e ett Figure 2
Lemma 3.1.
Let e i , . . . , e i m be the edges and let t j , . . . , t j m be the ideal triangles of T intersecting γ inthe cyclic order induced by the orientation of γ so that e i k is the common edge of t j k − and t j k for each k ∈{ , . . . , m } . Then up to a conjugation by an element of
P SL (2 , R ) ,ρ ([ γ ]) = ± S ( e i ) R ( t j ) S ( e i ) R ( t j ) . . . S ( e i m ) R ( t j m ) . Proof.
The proof is parallel to that of Lemma 3 in [3]. The idea is to keep track of the image of the unit tangentvector ∂∂y at i ∈ H under ρ ([ γ ]) . The contributions of each edge e and of each ideal triangle t intersecting γ to ρ ([ γ ]) are respectively ± S ( e ) and ± R ( t ) . See also Exercise 8.5-8.7 and 10.14 in [2].7or each puncture v of Σ g,n , let γ v be the simple closed curve going counterclockwise around v once. ByLemma 3.1, the image of γ v is up to conjugation ρ ([ γ v ]) = ± (cid:20) ψ v,(cid:15) ( λ )0 1 (cid:21) , where ψ v,(cid:15) is a rational function of λ depending on (cid:15). Therefore, ρ is type-preserving if and only if ψ v,(cid:15) ( λ ) (cid:54) = 0 for all punctures v. The following proposition gives a more precise description of this rational function in Theorem2.2.
Proposition 3.2 (Kashaev) . Let ( λ, (cid:15) ) ∈ R E> × {± } T , let V be the set of punctures of Σ g,n and let ψ (cid:15) be therational function defined by ψ (cid:15) = (cid:89) v ∈ V ψ v,(cid:15) . Then ( λ, (cid:15) ) defines a type-preserving representation if and only if ψ (cid:15) ( λ ) (cid:54) = 0 . The following theorem provides a more direct way to calculate the absolute values of the traces of closed curvesusing the λ -lengths, which is first found by Sun and the author in [28]. We include a proof here for the reader’sconvenience. For each ideal triangle t intersecting γ, let e be the edge of t at which γ enters, let e be the edge of t at which γ leaves and let e be the other edge of t. See Figure 2. Define M ( t ) = (cid:20) λ ( e ) (cid:15) ( t ) λ ( e )0 λ ( e ) (cid:21) if γ makes a left turn in t , and define M ( t ) = (cid:20) λ ( e ) 0 (cid:15) ( t ) λ ( e ) λ ( e ) (cid:21) if γ makes a right turn in t. Theorem 3.3.
For an immersed closed normal curve γ on Σ g,n , let e i , . . . , e i m be the edges and let t j , . . . , t j m be the ideal triangles of T intersecting γ in the cyclic order following the orientation of γ so that e i k is the commonedge of t j k − and t j k for each k ∈ { , . . . , m } . Then (cid:12)(cid:12) trρ ([ γ ]) (cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) M ( t j ) . . . M ( t j m ) (cid:1)(cid:12)(cid:12) λ ( e i ) . . . λ ( e i m ) . (3.1) Proof.
For each ideal triangle t and an edge e of t, let e (cid:48) and e (cid:48)(cid:48) be the other two edges of t so that the cyclic order e (cid:55)→ e (cid:48) (cid:55)→ e (cid:48)(cid:48) (cid:55)→ e coincides with the one induced by the orientation of Σ g,n . Define the matrix S ( t, e ) = (cid:113) λ ( e (cid:48)(cid:48) ) λ ( e (cid:48) ) (cid:113) λ ( e (cid:48) ) λ ( e (cid:48)(cid:48) ) . Then S ( e i k ) = S ( t j k − , e i k ) S ( t j k , e i k ) (3.2)for each k ∈ { , . . . , m } , where as a convention t j − = t j m . A case by case calculation shows that S ( t j k , e i k ) R ( t j k ) S ( t j k , e i k +1 ) = M ( t j k ) (cid:112) λ ( e i k ) λ ( e i k +1 ) (3.3)for each k ∈ { , . . . , m } , where as a convention e i m +1 = e i . By Lemma 3.1, (3.2), (3.3) and the fact that tr ( AB ) = tr ( BA ) for any two matrices A and B, we have (cid:12)(cid:12) trρ ([ γ ]) (cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) S ( e i ) R ( t j ) . . . S ( e i m ) R ( t j m ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) S ( t j , e i ) R ( t j ) S ( t j , e i ) . . . S ( t j m , e i m ) R ( t j m ) S ( t j m , e i ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) M ( t j ) . . . M ( t j m ) (cid:1)(cid:12)(cid:12) λ ( e i ) . . . λ ( e i m ) . emark . Formula (3.1) is first obtained by Roger-Yang [26] for decorated hyperbolic surfaces, i.e., discreteand faithful decorated representations, using the skein relations, where their formula includes both the traces ofclosed geodesics and the λ -lengths of geodesics arcs connecting the punctures. It is interesting to know whetherthere is a similar formula for the λ -lengths of arcs for the non-Fuchsian decorated representations.As a consequence of Theorem 3.3, we have the following Theorem 3.5. (1) For every non-Fuchsian type-preserving representation ρ : Σ g,n → P SL (2 , R ) , there exists aFuchsian type-preserving representation ρ (cid:48) such that (cid:12)(cid:12) trρ ([ γ ]) (cid:12)(cid:12) (cid:54) (cid:12)(cid:12) trρ (cid:48) ([ γ ]) (cid:12)(cid:12) for each [ γ ] ∈ π (Σ g,n ) , and the strict equality holds for at least one γ. (2) Conversely, for almost every Fuchsian type-preserving representation ρ (cid:48) : Σ g,n → P SL (2 , R ) and for each k with | k | < g − n and M k (Σ g,n ) (cid:54) = ∅ , there exists a type-preserving representation ρ with e ( ρ ) = k suchthat (cid:12)(cid:12) trρ ([ γ ]) (cid:12)(cid:12) (cid:54) (cid:12)(cid:12) trρ (cid:48) ([ γ ]) (cid:12)(cid:12) for each [ γ ] ∈ π (Σ g,n ) , and the strict equality holds for at least one γ. Proof.
For (1), by Theorem 2.1, there exists a ρ -admissible ideal triangulation T . Choose arbitrarily a decoration d of ρ, and let ( ρ (cid:48) , d (cid:48) ) be the decorated representation that has the same λ -lengths of ( ρ, d ) and has the positive signsfor all the ideal triangles. By (2.1) and Goldman’s result in [9], ρ (cid:48) is Fuchsian. Applying Formula (3.1) to | trρ ([ γ ]) | and | trρ (cid:48) ([ γ ]) | , we see that they have the same summands with different coefficients ± , and the coefficients for thelater are all positive. Since each summand is a product of the λ -lengths, which is positive, the inequality follows.Since ρ is non-Fuchsian, by (2.1), there must be an ideal triangle t that has negative sign in ( ρ, d ) . Therefore, if γ intersects t, then some of the summands in the expression of | trρ ([ γ ]) | has negative coefficients, and the inequalityfor γ is strict.For (2), choose arbitrarily an ideal triangulation T of Σ g,n , and let T be the set of ideal triangles of T . ByTheorems 2.1, 2.2 and 2.4, if M k (Σ g,n ) (cid:54) = ∅ , then there exists (cid:15) ∈ {± } T such that (cid:80) t ∈ T (cid:15) ( t ) = 2 k and thesubset R ( T , (cid:15) ) is homeomorphic via the lengths coordinate to a full measure open subset Ω( T , (cid:15) ) of R E> . For each λ ∈ Ω( T , (cid:15) ) , let ( ρ, d ) be the decorated representation determined by ( λ, (cid:15) ) . Then e ( ρ ) = k. On the other hand, R E> is identified with the decorated Teichm¨uller space via the lengths coordinate, hence λ determines a Fuchsiantype-preserving representation ρ (cid:48) . By the same argument in (1), the inequality holds for ρ and ρ (cid:48) , and is strict for γ intersecting the ideal triangles t with (cid:15) ( t ) = − . Remark . It is very interesting to know if (2) holds for every Fuchsian type-preserving representation. Thisamounts to ask wether (cid:91) T ,(cid:15) Ω( T , (cid:15) ) = R E> , where the union is over all the ideal triangulations T of Σ g,n and all (cid:15) that gives the right relative Euler class. A tetrahedral triangulation of the four-punctured sphere Σ , is an ideal triangulation of Σ , that is combina-torially equivalent to the boundary of an Euclidean tetrahedron (Figure 3(a)). A pair of edges of a tetrahedraltriangulation are called opposite if they are opposite edges of the tetrahedron. Let v , . . . , v be the four puncturesof Σ , . In the rest of this paper, for each tetrahedral triangulation T , we will let t i be the unique ideal triangleof T disjoint from the puncture v i and let e ij be the unique edge of T connecting the punctures v i and v j . Werespectively denote by x the pair of opposite edges { e , e } , by y the pair { e , e } and by z the pair { e , e } . See Figure 3 (b).A non-peripheral simple closed curve on Σ , is distinguished in a tetrahedral triangulation T if it is disjointfrom a pair of opposite edges of T and intersects each of the other four edges at exactly one point. In eachtetrahedral triangulation, there are exactly three distinguished simple closed curves. We respectively denote by X,Y and Z the distinguished simple closed curves disjoint from the pair of opposite edges x, y and z. See Figure9 zx y z xyee e e e ev vv v (a) (b) v vv v Figure 3 zxx yy z
X Y Z
Figure 44. The curves
X, Y and Z mutually intersect at exactly two points. On the other hand, for each triple of simpleclosed curves that mutually intersect at two points, there is a unique tetrahedral triangulation in which these threecurves are distinguished. In particular, each non-peripheral simple closed curve on Σ , is distinguished in sometetrahedral triangulation. Note that being the X -, Y - or Z -curve is independent of the choice of the tetrahedraltriangulation, since, for example, the curve X always separates { v , v } from { v , v } . In the rest of this paper,we will call a simple closed curve an X - (resp. Y - or Z -) curve if it is disjoint from the pair of opposite edges x (resp. y or z ) of some tetrahedral triangulation. In this way, we get a tri-coloring of the set of non-peripheralsimple closed curves on Σ , . A simultaneous diagonal switch at a pair of opposite edges of T is an operation that simultaneously doesdiagonal switches at this pair of edges. See Figure 5 (a). Denote respectively by S x , S y and S z the simultaneousdiagonal switches at the pair of opposite edges x, y and z. Then S x (reps. S y and S z ) changes the X - (resp. Y -and Z -) curve and leaves the other two distinguished simple closed curves unchanged. See Figure 5 (b). z z (cid:1) z Z (cid:1) Z S z z (cid:1) z z (cid:1) z S z z (cid:1) (a) (b) Figure 5The relationship between tetrahedral triangulations, simultaneous diagonal switches and non-peripheral simpleclosed curves can be described by (the dual of) the Farey diagram. Recall that the Farey diagram F is an idealtriangulation of H whose vertices are the extended rational numbers Q ∪ {∞} ⊂ ∂ H , and the dual Fereydiagram F ∗ is a countably infinite trivalent tree properly embedded in H . Each vertex of F corresponds to a10on-peripheral simple closed curve on Σ , , each edge of F connects two vertices corresponding to two simpleclosed curves that intersect at exactly two points and each ideal triangle of F corresponds to a triple of simpleclosed curves mutually intersecting at two points. (See e.g. [20].) Therefore, each vertex of the dual graph F ∗ corresponds to a tetrahedral triangulation of Σ , , each edge of F ∗ corresponds to a simultaneous diagonal switchand each connected component of H \ F ∗ corresponds to a non-peripheral simple closed curves on Σ , . SeeFigure 6. Since F ∗ is connected, any tetrahedral triangulation can be obtained from another by doing a finitelysequence of simultaneous diagonal switches. z Z Z (cid:1)
Z Z (cid:1) S (cid:1) (cid:1) ƬƬ ƬƬ z S Figure 6We close up this section by showing the relationship between simultaneous diagonal switches and the mappingclasses of Σ , . Proposition 4.1.
A composition of an even number of simultaneous diagonal switches determines an elementof
M od (Σ , ) . Conversely, any element of
M od (Σ , ) is determined by a composition of an even number ofsimultaneous diagonal switches.Proof. Let T be a tetrahedra triangulation of Σ , . We write φ = S (cid:48) S if φ is the self-diffeomorphism of Σ , suchthat the tetrahedral triangulation φ ( T ) is obtained from T by doing the simultaneous diagonal switch S followedby the simultaneous diagonal switch S (cid:48) . Then D X = S z S y and D Y = S x S z . See Figure 7. Similarly, we have zy 𝐷 S SX xz 𝐷 S SY XY Figure 7 S y S x = D Z , S y S z = D − X , S x S z = D − Y and S x S y = D − X . Thus, any composition of an even number ofsimultaneous diagonal switches determines an element of
M od (Σ , ) . { x, y, z } of paris of opposite edges of T asfollows. Since each puncture v of Σ , is adjacent to three edges e, e (cid:48) and e (cid:48)(cid:48) with e ∈ x, e (cid:48) ∈ y and e (cid:48)(cid:48) ∈ z, the orientation of Σ , induces a cyclic order on the set { e, e (cid:48) , e (cid:48)(cid:48) } around v, inducing a cyclic order on the set { x, y, z } . It is easy to check that this cyclic order is independent of the choose of v, hence is well defined. Wecall the sign of a tetrahedral triangulation T positive if the cyclic order x (cid:55)→ y (cid:55)→ z (cid:55)→ x coincides with the oneinduced from the orientation, and negative if otherwise. It easy to see that a simultaneous diagonal switch changesthe sign of T , and an orientation preserving self-diffeomorphism of Σ , preserves the sign of T . Since the dualFarey diagram F ∗ is a connected tree, for any self-diffeomorphism φ of Σ , , up to redundancy there is a uniquepath of F ∗ connecting the vertices T and φ ( T ) . Since T and φ ( T ) have the same sign, the path consists of aneven number of edges, corresponding to an even number of simultaneous diagonal switche S , . . . , S m . Then φ = φ k ◦ · · · ◦ φ , where φ k = S k S k − . Let ρ be a type-preserving representation of π (Σ , ) and let d be a decoration of ρ. Suppose T is a ρ -admissibletetrahedral triangulation of Σ , , E and T respectively are the sets of edges and ideal triangles of T , and ( λ, (cid:15) ) ∈ R E> × {± } T is the lengths coordinate of [( ρ, d )] ∈ M d ± (Σ , ) . Let v , . . . , v be the punters of Σ , , let t i bethe ideal triangle of T disjoint from v i and let e ij be the edge of T connecting the punctures v i and v j . Define thequantities λ ( x ) = λ ( e ) λ ( e ) , λ ( y ) = λ ( e ) λ ( e ) and λ ( z ) = λ ( e ) λ ( e ) . The quantities λ ( x ) , λ ( y ) and λ ( z ) will play a central role in the rest of this paper. Suppose e ( ρ ) = 1 . Then by (2.1), there is exactly one ideal triangle, say t , such that (cid:15) ( t ) = − and (cid:15) ( t i ) = 1 for i (cid:54) = 1 . As a direct consequence of Lemma 3.1 and Theorem 3.3, we have the following lemmas.
Lemma 5.1.
Let γ i be the simple closed curve going counterclockwise around the puncture v i once. Then up toconjugation, the ρ -image of the peripheral element [ γ ] ∈ π (Σ , ) is ± (cid:20) λ ( x ) + λ ( y ) + λ ( z )0 1 (cid:21) , and the ρ -image of the other peripheral elements [ γ ] , [ γ ] and [ γ ] are respectively ± (cid:20) λ ( y ) + λ ( z ) − λ ( x )0 1 (cid:21) , ± (cid:20) λ ( x ) + λ ( z ) − λ ( y )0 1 (cid:21) and ± (cid:20) λ ( x ) + λ ( y ) − λ ( z )0 1 (cid:21) . Lemma 5.2. (1) The absolute values of the traces of the distinguished simple closed curves
X, Y and Z of T canbe calculated by (cid:12)(cid:12) trρ ([ X ]) (cid:12)(cid:12) = (cid:12)(cid:12) λ ( y ) + λ ( z ) − λ ( x ) (cid:12)(cid:12) λ ( y ) λ ( z ) , (cid:12)(cid:12) trρ ([ Y ]) (cid:12)(cid:12) = (cid:12)(cid:12) λ ( x ) + λ ( z ) − λ ( y ) (cid:12)(cid:12) λ ( x ) λ ( z ) and (cid:12)(cid:12) trρ ([ Z ]) (cid:12)(cid:12) = (cid:12)(cid:12) λ ( x ) + λ ( y ) − λ ( z ) (cid:12)(cid:12) λ ( x ) λ ( y ) . (5.1) (2) The right hand sides of the equations in (5.1) are strictly greater than if and only if λ ( x ) , λ ( y ) and λ ( z ) satisfy one of the following inequalities λ ( x ) > λ ( y ) + λ ( z ) ,λ ( y ) > λ ( x ) + λ ( z ) or λ ( z ) > λ ( x ) + λ ( y ) . (5.2)12ote that reversing the directions of the inequalities in (5.2), we get the triangular inequality. The idea of theproof of (2) is that if we regard the quantities λ ( x ) , λ ( y ) and λ ( z ) as the edge lengths of a Euclidean triangle, thenthe right hand sides of (5.1) are twice of the cosine of the corresponding inner angles. The next lemma shows therule of the change of the quantities λ ( x ) , λ ( y ) and λ ( z ) under a simultaneous diagonal switch. Lemma 5.3.
Suppose T (cid:48) is a tetrahedral triangulation of Σ , . If T (cid:48) is ρ -admissible, then let λ (cid:48) be the λ -lengthsof ( ρ, d ) in T (cid:48) , and let λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) be the corresponding quantities.(1) If T (cid:48) is obtained from T by doing S x , then T (cid:48) is ρ -admissible if and only if λ ( y ) (cid:54) = λ ( z ) . In the case that T (cid:48) is ρ -admissible, λ (cid:48) ( y ) = λ ( y ) , λ (cid:48) ( z ) = λ ( z ) and λ (cid:48) ( x ) = (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) λ ( x ) . (2) If T (cid:48) is obtained from T by doing S y , then T (cid:48) is ρ -admissible if and only if λ ( x ) (cid:54) = λ ( z ) . In the case that T (cid:48) is ρ -admissible, λ (cid:48) ( x ) = λ ( x ) , λ (cid:48) ( z ) = λ ( z ) and λ (cid:48) ( y ) = (cid:12)(cid:12) λ ( z ) − λ ( x ) (cid:12)(cid:12) λ ( y ) . (3) If T (cid:48) is obtained from T by doing S z , then T (cid:48) is ρ -admissible if and only if λ ( x ) (cid:54) = λ ( y ) . In the case that T (cid:48) is ρ -admissible, λ (cid:48) ( x ) = λ ( x ) , λ (cid:48) ( y ) = λ ( y ) and λ (cid:48) ( z ) = (cid:12)(cid:12) λ ( x ) − λ ( y ) (cid:12)(cid:12) λ ( z ) . Proof.
For (1), we have that the edge e is adjacent to the ideal triangle t and t with (cid:15) ( t ) = (cid:15) ( t ) and e is adjacent to the ideal triangles t and t with (cid:15) ( t ) (cid:54) = (cid:15) ( t ) . Let e (cid:48) and e (cid:48) respectively be the edges of T (cid:48) obtained from diagonal switches at e and e , i.e., e (cid:48) is the edge of T (cid:48) connecting the punctures v and v and e (cid:48) is the edge of T (cid:48) connecting the punctures v and v . By Proposition 2.3, T (cid:48) is ρ -admissible if andonly if λ ( e ) λ ( e ) (cid:54) = λ ( e ) λ ( e ) , i.e., λ ( y ) (cid:54) = λ ( z ) . By Proposition 2.3 again, if T (cid:48) is ρ -admissible, then λ (cid:48) ( e ij ) = λ ( e ij ) for { i, j } (cid:54) = { , } or { , } , and λ (cid:48) ( e (cid:48) ) = (cid:12)(cid:12) λ ( e ) λ ( e ) − λ ( e ) λ ( e ) (cid:12)(cid:12) λ ( e ) and λ (cid:48) ( e (cid:48) ) = λ ( e ) λ ( e ) + λ ( e ) λ ( e ) λ ( e ) . Therefore, λ (cid:48) ( y ) = λ ( y ) , λ (cid:48) ( z ) = λ ( z ) and λ (cid:48) ( x ) = λ (cid:48) ( e (cid:48) ) λ (cid:48) ( e (cid:48) ) = (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) /λ ( x ) . The proofs of (2) and (3) are the similar.A consequence of Lemma 5.3 is that the inequalities in (5.2) are persevered by the simultaneous diagonalswitches.
Lemma 5.4.
Suppose T (cid:48) is a ρ -admissible tetrahedral triangulation of Σ , obtained from T by doing a simulta-neous diagonal switch. Let λ (cid:48) be the λ -lengths of ( ρ, d ) in T (cid:48) , and let λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) be the correspondingquantities. Then λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) satisfy one of the inequalities in (5.2) if and only if λ ( x ) , λ ( y ) and λ ( z ) do.Proof. Without lost of generality, we assume that T (cid:48) is obtained from T by doing S x . If λ ( x ) > λ ( y ) + λ ( z ) , thenby Lemma 5.3, λ (cid:48) ( x ) = (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) λ ( x ) < (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) λ ( y ) + λ ( z ) = (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) = (cid:12)(cid:12) λ (cid:48) ( y ) − λ (cid:48) ( z ) (cid:12)(cid:12) . λ (cid:48) ( y ) > λ (cid:48) ( x ) + λ (cid:48) ( z ) or λ (cid:48) ( z ) > λ (cid:48) ( x ) + λ (cid:48) ( y ) . On the other hand, if either λ ( y ) > λ ( x ) + λ ( z ) or λ ( z ) > λ ( x ) + λ ( y ) , i.e., λ ( x ) < (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) , then by Lemma 5.3, λ (cid:48) ( x ) = (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) λ ( x ) > (cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) λ ( y ) − λ ( z ) (cid:12)(cid:12) = λ ( y ) + λ ( z ) = λ (cid:48) ( y ) + λ (cid:48) ( z ) . Another consequence of Lemma 5.3 is the following
Proposition 5.5.
There are uncountably many [ ρ ] ∈ M ± (Σ , ) such that all the tetrahedral triangulations of Σ , are ρ -admissible.Proof. Suppose ρ is a typer-preserving representation of π (Σ , ) with e ( ρ ) = 1 , and d is a decoration of ρ. Let T be a ρ -admissible tetrahedral triangulation of Σ , and let ( λ, (cid:15) ) be the lengths coordinate of ( ρ, d ) in T . Recallthat there is a ono-to-one correspondence between the tetrahedral triangulations of Σ , and the vertices of the dualFarey diagram F ∗ , which is a countably infinity tree. Therefore, for each tetrahedral triangulation T (cid:48) , there is up toredundancy a unique path in F ∗ connecting T and T (cid:48) , which corresponds to a sequence { S i } ni =1 of simultaneousdiagonal switches. Let T = T , and for each i ∈ { , . . . , n } , let T i be the tetrahedral triangulation obtained from T i − by doing S i . Suppose T i is ρ -admissible for some i ∈ { , . . . , n } , and suppose λ i is the λ -lengths of ( ρ, d ) in T i . Then by Lemma 5.3, T i +1 is ρ -admissible if and only if the Laurent polynomial λ i ( y ) − λ i ( z ) λ i ( x ) (cid:54) = 0 . An inductionin i shows that T (cid:48) = T n is ρ -admissible if and only if certain Laurent polynomial L T (cid:48) ( λ ( x ) , λ ( y ) , λ ( z )) (cid:54) = 0 . Theset of zeros Z T (cid:48) of L T (cid:48) is a Zariski-closed subset of R > . In particular, the Lebesgue measure m ( Z T (cid:48) ) = 0 . Since F ∗ is a countably infinity tree, there are in total countably many tetrahedral triangulations T of Σ , , andhence m ( (cid:83) T Z T ) = 0 . Therefore, the set C = R > \ (cid:83) T Z T has a full measure in R > . In particular, C containsuncountable many points.Now each ( a, b, c ) ∈ C with a + b (cid:54) = c, a + c (cid:54) = b and b + c (cid:54) = a determines a type-preserving representation ρ as follows. Take a tetrahedral triangulation T of Σ , , and let E and T respectively be the set of edges andideal triangles of T . Choose (cid:15) ∈ {± } T so that (cid:80) t ∈ T (cid:15) ( t ) = 2 , and define λ ∈ R E> by λ ( e ) = λ ( e ) = a ,λ ( e ) = λ ( e ) = b and λ ( e ) = λ ( e ) = c . Then λ ( x ) = a, λ ( y ) = b and λ ( z )) = c. By Theorem 2.2,Proposition 3.2 and Lemma 5.1, ( λ, (cid:15) ) determines a decorated representation ( ρ, d ) up to conjugation. In particular,by Lemma 5.1, ρ is type-preserving. By (2.1), the relative Euler class e ( ρ ) = 1 . Finally, since ( λ ( x ) , λ ( y ) , λ ( z )) ∈C , the Laurent polynomial L T (cid:48) ( λ ( x ) , λ ( y ) , λ ( z )) (cid:54) = 0 for all tetrahedral triangulation T (cid:48) . As a consequence, allthe tetrahedral triangulations are ρ -admissible.By symmetry, there are also uncountably many type-preserving representations ρ with e ( ρ ) = − such that allthe tetrahedral triangulations of Σ , are ρ -admissible. Proof of Theorem 1.1.
Let T be a tetrahedral triangulation of Σ , and let C be the full measure subset of R > constructed in the proof of Proposition 5.5. Then each ( a, b, c ) ∈ C satisfying one of the following identities a > b + c, b > a + c or c > a + b determines a decorated representation ( ρ, d ) with e ( ρ ) = ± such that all thetetrahedral triangulations of Σ , are ρ -admissible. Since elementary representation have relative Euler class and e ( ρ ) = ± , ρ is non-elementary. For each tetrahedral triangulation T (cid:48) , let λ (cid:48) be the λ -lengths of ( ρ, d ) in T (cid:48) . Since T (cid:48) can by obtained from T by doing a sequence of simultaneous diagonal switches, by Lemma 5.4, the quantities λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) satisfy one of the inequalities in (5.2). By Lemma 5.2, the traces of the distinguished simpleclosed curves X, Y and Z in T (cid:48) are strictly greater than in the absolute value. Since each simple closed curve γ is distinguished in some tetrahedral triangulation T (cid:48) , we have (cid:12)(cid:12) trρ ([ γ ]) (cid:12)(cid:12) > . Suppose e ( ρ ) = 0 . Then by (2.1), there are exactly two ideal triangles having the positive sign and two having thenegative sign. Without loss of generality, we assume that (cid:15) ( t ) = (cid:15) ( t ) = − and (cid:15) ( t ) = (cid:15) ( t ) = 1 . Note thatunder this assumption, the edges e and e in the pair x are adjacent to ideal triangles having the same sign, andas will be seen later, the X -curves will play a different role than the Y - and Z - curves do. As a direct consequenceof Lemma 3.1, we have the following 14 emma 5.6. Let γ i be the simple closed going counterclockwise around the puncture v i . Then up to conjugation,the ρ -image of the peripheral elements [ γ ] and [ γ ] of π (Σ , ) are ± (cid:20) λ ( y ) + λ ( z ) − λ ( x )0 1 (cid:21) , and the ρ -image of the other two peripheral elements [ γ ] and [ γ ] are ± (cid:20) λ ( x ) − λ ( y ) − λ ( z )0 1 (cid:21) . Lemma 5.7. (1) The absolute values of the traces of the distinguished simple closed curves
X, Y and Z of T canbe calculated by (cid:12)(cid:12) trρ ([ X ]) (cid:12)(cid:12) = (cid:12)(cid:12) λ ( x ) + λ ( y ) + λ ( z ) − λ ( x ) λ ( y ) − λ ( x ) λ ( z ) (cid:12)(cid:12) λ ( y ) λ ( z ) , (cid:12)(cid:12) trρ ([ Y ]) (cid:12)(cid:12) = λ ( x ) + λ ( y ) + λ ( z ) + 2 λ ( y ) λ ( z ) − λ ( x ) λ ( y ) λ ( x ) λ ( z ) and (cid:12)(cid:12) trρ ([ Z ]) (cid:12)(cid:12) = λ ( x ) + λ ( y ) + λ ( z ) + 2 λ ( y ) λ ( z ) − λ ( x ) λ ( z ) λ ( x ) λ ( y ) . (5.3) (2) The right hand sides of the last two equations in (5.3) are always strictly greater than , whereas the righthand side of the first equation is less than or equal to if and only if λ ( x ) , λ ( y ) and λ ( z ) satisfy the followinginequalities (cid:112) λ ( x ) (cid:54) (cid:112) λ ( y ) + (cid:112) λ ( z ) , (cid:112) λ ( y ) (cid:54) (cid:112) λ ( x ) + (cid:112) λ ( z ) and (cid:112) λ ( z ) (cid:54) (cid:112) λ ( x ) + (cid:112) λ ( y ) . (5.4) Proof. (1) is a direct consequence of Theorem 3.3. For (2), since ρ is type-preserving, by Theorem 2.2, Proposition3.2 and Lemma 5.6, λ ( x ) − λ ( y ) − λ ( z ) (cid:54) = 0 . Therefore, the right hand side of the second equation of (5.3) equals (cid:0) λ ( x ) − λ ( y ) − λ ( z ) (cid:1) λ ( x ) λ ( z ) + 2 > , and the right hand side of the third equation equals (cid:0) λ ( x ) − λ ( y ) − λ ( z ) (cid:1) λ ( x ) λ ( y ) + 2 > . In thecase that λ ( x ) + λ ( y ) + λ ( z ) − λ ( x ) λ ( y ) − λ ( x ) λ ( z ) (cid:62) , the right hand side of the first equation of (5.3)equals (cid:0) λ ( x ) − λ ( y ) − λ ( z ) (cid:1) λ ( y ) λ ( z ) − > − . The quantity also equals − ( λ ( x ) + λ ( y ) + λ ( z ) )( λ ( x ) + λ ( y ) − λ ( z ) )( λ ( x ) + λ ( z ) − λ ( y ) )( λ ( y ) + λ ( z ) − λ ( x ) ) λ ( y ) λ ( z ) , which is less than or equal to if and only if the equalities in (5.4) are satisfied. For the case that λ ( x ) + λ ( y ) + λ ( z ) − λ ( x ) λ ( y ) − λ ( x ) λ ( z ) (cid:54) , the proof is similar.The next lemma shows the rule of the change of the quantities λ ( x ) , λ ( y ) and λ ( z ) under a simultaneousdiagonal switch. Lemma 5.8.
Suppose T (cid:48) is a tetrahedral triangulation of Σ , . If T (cid:48) is ρ -admissible, then let λ (cid:48) be the λ -lengthsof ( ρ, d ) in T (cid:48) , and let λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) be the corresponding quantities.(1) If T (cid:48) is obtained from T by doing S x , then T (cid:48) is ρ -admissible. In this case, λ (cid:48) ( y ) = λ ( y ) , λ (cid:48) ( z ) = λ ( z ) and λ (cid:48) ( x ) = (cid:0) λ ( y ) + λ ( z ) (cid:1) λ ( x ) . (2) If T (cid:48) is obtained from T by doing S y , then T (cid:48) is ρ -admissible if and only if λ ( x ) (cid:54) = λ ( z ) . In the case that T (cid:48) is ρ -admissible, λ (cid:48) ( x ) = λ ( x ) , λ (cid:48) ( z ) = λ ( z ) and λ (cid:48) ( y ) = (cid:0) λ ( z ) − λ ( x ) (cid:1) λ ( y ) .
3) If T (cid:48) is obtained from T by doing S z , then T (cid:48) is ρ -admissible if and only if λ ( x ) (cid:54) = λ ( z ) . In the case that T is ρ -admissible, λ (cid:48) ( x ) = λ ( x ) , λ (cid:48) ( y ) = λ ( y ) and λ (cid:48) ( z ) = (cid:0) λ ( x ) − λ ( y ) (cid:1) λ ( z ) . Proof.
This is a consequence of Proposition 2.3, and the proof is similar to that of Lemma 5.3.
Proof of Theorem 1.2.
Let ρ be a type-preserving representation of π (Σ , ) with relative Euler class e ( ρ ) = 0 , and choose arbitrarily a decoration d of ρ. Let T be a tetrahedral triangulation of Σ , . If T is not ρ -admissible,then there is an edge e of T that is not ρ -admissible, and the element of π (Σ , ) represented by the distinguishedsimple closed curve in T disjoint from e is sent by ρ to a parabolic element of P SL (2 , R ) . If T is ρ -admissible, thenwe let ( λ, (cid:15) ) be the lengths coordinate of ( ρ, d ) in T . If the quantities λ ( x ) , λ ( y ) and λ ( z ) satisfy the inequalitiesin (5.4), then by Lemma 5.2, the element of π (Σ , ) represented by one of the distinguished simple closed curves X, Y and Z is sent by ρ to either an elliptic or a parabolic element of P SL (2 , R ) . Therefore, to prove the theorem,it suffices to find a tetrahedral triangulation T (cid:48) of Σ , such that either T (cid:48) is not ρ -admissible or T (cid:48) is ρ -admissiblewith the quantities λ (cid:48) ( x ) , λ (cid:48) ( y ) and λ (cid:48) ( z ) satisfying the inequalities in (5.4). Our strategy of finding T (cid:48) is toconstruct a sequence of tetrahedral triangulations {T n } Nn =1 with T N = T (cid:48) by the following Trace Reduction Algorithm:
Let T = T and suppose that T n is obtained. If T n is not ρ -admissible, then westop. If T n is ρ -admissible, then we let ( λ n , (cid:15) n ) be the lengths coordinate of ( ρ, d ) in T n . If λ n ( x ) , λ n ( y ) and λ n ( z ) satisfy the inequalities in (5.4), then we stop. If otherwise, then there is a unique maximum among λ n ( x ) ,λ n ( y ) and λ n ( z ) , since other wise the inequalities (5.4) are satisfied. Suppose { e ij , e kl } is the pair of oppositeedges of T n such that λ ( e ij ) λ ( e kl ) equals the maximum of λ n ( x ) , λ n ( y ) and λ n ( z ) . Then we let T n +1 be thetetrahedral triangulation obtained from T n by doing a simultaneous diagonal switch at e ij and e kl . By Lemma 5.9 below, the algorithm stops at some T N . Lemma 5.9.
The Trace Reduction Algorithm stops in finitely many steps.Proof.
For each n, let t i be the ideal triangle of T n disjoint from the puncture v i , and let e ij be the edge of T n connecting the punctures v i and v j . Without loss of generality, we assume in T that (cid:15) ( t ) = (cid:15) ( t ) = − and (cid:15) ( t ) = (cid:15) ( t ) = 1 . Then by Proposition 2.3, (cid:15) n ( t ) = (cid:15) n ( t ) and (cid:15) n ( t ) = (cid:15) n ( t ) for each T n . For each n, we let a n = λ n ( x ) λ n ( x )+ λ n ( y )+ λ ( z ) , b n = λ n ( y ) λ n ( x )+ λ n ( y )+ λ n ( z ) and c n = λ n ( z ) λ n ( x )+ λ n ( y )+ λ n ( z ) , and let k n = max (cid:8) √ a n − (cid:112) b n − √ c n , (cid:112) b n − √ a n − √ c n , √ c n − √ a n − (cid:112) b n (cid:9) . Then λ n ( x ) , λ n ( y ) and λ n ( z ) satisfy the inequalities in (5.4) if and only if k n (cid:54) . Assume that the sequence {T n } is infinite, i.e., k n > for all n > . Then we will find a contradiction by thefollowing three steps. In Step I we show that k n is decreasing in n by considering two mutually complementarycases, where in one of them (Case 1) the gap k n − k n +1 is bounded below by the minimum of a n , b n and c n . InStep II we show that there must be a infinite subsequence {T n i } of {T n } such that each T n i is of Case 1 of Step I,and in Step III we show that for i large enough, min { a n i , b n i , c n i } is increasing. The three steps together implythat k n < for some n large enough, which is a contradiction.Step I: We show that k n is decreasing in n. There are the following two cases to verify.Case 1: √ a n − √ b n − √ c n > . In this case, by Lemma 5.8, (cid:0) a n +1 , b n +1 , c n +1 (cid:1) = (cid:16) b n + c n , a n b n b n + c n , a n c n b n + c n (cid:17) . (5.5)Without loss of generality, we assume that b n > c n . Then b n +1 is the largest among a n +1 , b n +1 and c n +1 . By adirect calculation and that √ a n > √ b n + √ c n , we have k n − k n +1 = (cid:0) √ a n − (cid:112) b n − √ c n (cid:1) − (cid:0)(cid:112) b n +1 − √ a n +1 − √ c n +1 (cid:1) > c n √ b n + c n > . a n + b n + c n = 1 and a n > , we have √ b n + c n < , and hence k n − k n +1 > c n . Therefore,we have k n − k n +1 > { a n , b n , c n } . (5.6)Case 2: One of √ b n − √ a n − √ c n and √ c n − √ a n − √ b n is strictly greater than . In this case, we withoutloss of generality assume that √ b n − √ a n − √ c n > . Then by Lemma 5.8, (cid:0) a n +1 ,b n +1 , c n +1 (cid:1) = (cid:18) a n b n a n b n + b n c n + ( a n − c n ) , ( a n − c n ) a n b n + b n c n + ( a n − c n ) , b n c n a n b n + b n c n + ( a n − c n ) (cid:19) . (5.7)Without loss of generality, we assume that a n > c n . Then a n +1 is the largest among a n +1 , b n +1 and c n +1 . By adirect calculation and that b n = 1 − a n − c n , we have k n +1 k n = √ a n +1 − (cid:112) b n +1 − √ c n +1 √ b n − √ a n − √ c n = (cid:115) a n + c n − √ a n c n a n + c n − a n c n . From √ b n > √ a n + √ c n and a n + b n + c n = 1 , we have a n < , c n < , and hence √ a n c n > a n c n . As aconsequence, k n +1 /k n < . Step II: We show that there is an infinite subsequence {T n i } of {T n } such that ( a n i , b n i , c n i ) is in Case 1 ofStep I. We use contradiction. For each ( a n , b n , c n ) in Case 2 of Step I, let A n = max { λ n ( y ) , λ n ( z ) } and let B n = min { λ n ( y ) , λ n ( z ) } . Then √ A n > √ B n + (cid:112) λ n ( x ) . By Lemma 5.8, ( a n +1 , b n +1 , c n +1 ) is in Case 1 ofStep I if and only if λ n ( x ) > B n . Now suppose that there is an m ∈ N such that ( a m , b m , c m ) is in Case 2 of StepI and B n > λ n ( x ) for all n (cid:62) m. Then by Lemma 5.8, we have λ n +1 ( x ) = λ n ( x ) and (cid:112) B n +1 = B n − λ n ( x ) √ A n < B n − λ n ( x ) √ B n + (cid:112) λ n ( x ) = (cid:112) B n − (cid:112) λ n ( x ) for n (cid:62) m. By induction, λ n ( x ) = λ m ( x ) and √ B n < √ B m − ( n − m ) (cid:112) λ m ( x ) for all n > m, which isimpossible.Step III: We show that for i large enough, min { a n i , b n i , c n i } is increasing. In Figure 8 below, we let ∆ = (cid:8) ( a, b, c ) ∈ R > | a + b + c = 1 (cid:9) , and for each k let C k be the intersection of ∆ with the set (cid:8) ( a, b, c ) ∈ R > | max {√ a − √ b − √ c, √ b − √ a − √ c, √ c − √ a − √ b } = k (cid:9) . A direct calculation shows that C k ’s are partsof the concentric circles centered at ( , , ) with radii increasing in k, and that C is the inscribed circle of ∆ . InFigure 8 (a), let Q be the intersection of ∆ and the set (cid:8) ( a, b, c ) ∈ R > | ( b + c ) = ac (cid:9) . Then Q is a quadraticcurve in ∆ going through the points (1 , , and ( , , ) . Let the line segment P be the intersection of ∆ andthe plane (cid:8) ( a, b, c ) ∈ R | a = c (cid:9) . Then by (5.5), if ( a n , b n , c n ) is on Q with b n > c n , then ( a n +1 , b n +1 , c n +1 ) is on P. Denote by H the line segment connecting ( , , and ( , , ) , and by L the line segment connecting (1 , , and (0 , , . Let D be the region of ∆ bounded by Q, H and L, and let E be the region in ∆ bounded by P, H and L. In Figure 8 (b), let p be the intersection of Q and C k , let (cid:15) be the third coordinate of p, let L (cid:15) be theintersection of ∆ and the plane { ( a, b, c ) ∈ R | c = (cid:15) } , and let F be the region in ∆ bounded by C k , L (cid:15) , H and L. Note that F is a subset of D. Now consider the infinite subsequence {T n i } guaranteed by Step II such that ( a n i , b n i , c n i ) is in Case 1 ofStep I. By (5.6), there exists an i such that min { b n i , c n i } < (cid:15) for all i > i , since otherwise k n i +1 < for i large enough, and the algorithm stops. Without loss of generality, we assume that b n i > c n i , and we have thefollowing two claims.Claim 1: If ( a n , b n , c n ) ∈ D and b n > c n , then ( a n +1 , b n +1 , c n +1 ) ∈ E, b n +1 > c n +1 and min { a n +1 , b n +1 , c n +1 } > min { a n , b n , c n } . Indeed, in this case, c n = min { a n , b n , c n } . By (5.5), ( a n +1 , b n +1 , c n +1 ) ∈ E and c n +1 > c n . Furthermore,we have b n +1 > c n +1 , since otherwise ( a n +1 , b n +1 , c n +1 ) would be in the disk bounded by the circle C , i.e., k n +1 < and the algorithm stops. Therefore, c n +1 = min { a n +1 , b n +1 , c n +1 } and min { a n +1 , b n +1 , c n +1 } > min { a n , b n , c n } . (1,0,0)(0,0,1) (0,1,0)(1/2,0,1/2) (1/2,1/2,0) LHC P p ɛ C k D E
LH L (a) (b) F Figure 8Claim 2: For n > n , if ( a n , b n , c n ) ∈ E and b n > c n , then ( a n +1 , b n +1 , c n +1 ) ∈ D, b n +1 > c n +1 and min { a n +1 , b n +1 , c n +1 } > min { a n , b n , c n } . Indeed, in this case, c n = min { a n , b n , c n } . By (5.7), ( a n +1 , b n +1 , c n +1 ) is in the triangle above H, b n +1 > c n +1 and c n +1 > c n . Therefore, c n +1 = min { a n +1 , b n +1 , c n +1 } and min { a n +1 , b n +1 , c n +1 } > min { a n , b n , c n } . Furthermore, since n > n i , we have c n +1 < (cid:15), and by Step I, we have k n +1 < k . As a consequence, ( a n +1 , b n +1 , c n +1 ) ∈ F ⊂ D. Since the intersection ( , , ) of the quadratic curve Q and the circle C lieson the line determined by b = c, F lies on the right half of ∆ , and hence b n +1 > c n +1 . Since k n i < k and by assumption b n i > c n i and c n i < (cid:15), we have ( a n i , b n i , c n i ) ∈ F ⊂ D. By aninduction and Claims 1 and 2, we have for all m (cid:62) that ( a n i +2 m , b n i +2 m , c n i +2 m ) ∈ D with b n i +2 m >c n i +2 m and ( a n i +2 m +1 , b n i +2 m +1 , c n i +2 m +1 ) ∈ E with b n i +2 m +1 > c n i +2 m +1 , and hence for n > n i have min { a n +1 , b n +1 , c n +1 } > min { a n , b n , c n } . Similar to the relative Euler class ± case, we have Proposition 5.10.
There are uncountably many [ ρ ] ∈ M (Σ , ) such that all the tetrahedral triangulations of Σ , are ρ -admissible. For each such ρ, there is a simple closed curve γ on Σ , such that ρ ([ γ ]) is an ellipticelement in P SL (2 , R ) . Proof.
Since the functions in Lemma 5.8 are rational in λ ( x ) , λ ( y ) and λ ( z ) , the argument in the proof of Propo-sition 5.5 applies here and proves the first part. The second part is a result of the Trace Reduction Algorithm. M (Σ , ) We describe the connected component of the character space M (Σ , ) in this section. Recall that for a quadruple s of positive and negative signs, M sk (Σ , ) is the space of conjugacy classes of type-preserving representationswith relative Euler class k and signs of the punctures s. Let V = { v , . . . , v } be the set of punctures of Σ , . ThenTheorem 1.3 is equivalent to the following
Theorem 6.1. (1) For { i, j, k, l } = { , , , } , let s ij ∈ {± } V be defined by s ij ( v i ) = s ij ( v j ) = − and s ij ( v k ) = s ij ( v l ) = +1 . Then M (Σ , ) = (cid:97) { i,j }⊂{ ,..., } M s ij (Σ , ) .
2) For i ∈ { , . . . , } , let s i ∈ {± } V be defined by s i ( v i ) = − and s i ( v j ) = +1 for j (cid:54) = i, and let s + ∈ {± } V be defined by s + ( v i ) = 1 for all i ∈ { , . . . , } . Then M (Σ , ) = (cid:97) i =1 M s i (Σ , ) (cid:97) M s + (Σ , ) . (3) For i ∈ { , . . . , } , let s − i ∈ {± } V be defined by s − i ( v i ) = +1 and s − i ( v j ) = − for j (cid:54) = i, and let s − ∈ {± } V be defined by s − ( v i ) = − for all i ∈ { , . . . , } . Then M − (Σ , ) = (cid:97) i =1 M s − i − (Σ , ) (cid:97) M s − − (Σ , ) . (4) All the spaces M s ij (Σ , ) , M s i (Σ , ) , M s + (Σ , ) , M s − i − (Σ , ) and M s − − (Σ , ) are connected. Let T be a tetrahedral triangulation of Σ , . Recall that M T (Σ , ) is the space of conjugacy classes of type-preserving representations ρ such that T is ρ -admissible. By Theorem 2.1, M T (Σ , ) is a dense and open subsetof M (Σ , ) . Let E and T respectively be the sets of edges and ideal triangles of T , let t i ∈ T be the ideal triangledisjoint from the puncture v i and let e ij ∈ E be the edge connecting the punctures v i and v j . For λ ∈ R E> , let λ ( x ) = λ ( e ) λ ( e ) , λ ( y ) = λ ( e ) λ ( e ) and λ ( z ) = λ ( e ) λ ( e ) . We first show that the quantities λ ( x ) ,λ ( y ) and λ ( z ) parametrize the components of M T (Σ , ) . Lemma 6.2.
Let R E> be with the principle R V> -bundle structure given by ( µ · λ )( e ij ) = µ ( v i ) λ ( e ij ) µ ( v j ) , and let R > be with the principle R > -bundle structure defined by r · ( a, b, c ) = ( ra, rb, rc ) . Then the map φ : R E> → R > sending (cid:0) λ ( e ) , . . . , λ ( e ) (cid:1) to (cid:0) λ ( x ) , λ ( y ) , λ ( z ) (cid:1) induces a diffeomorphism φ ∗ : R E> / R V> → R > / R > . Proof.
Since φ ( µ · λ ) = (cid:81) i =1 µ ( v i ) · φ ( λ ) , φ ∗ is well defined, and since φ ( a , b , c , c , b , a ) = ( a, b, c ) for all ( a, b, c ) ∈ R > , φ ∗ is surjective. For the injectivity, we suppose that φ ( λ (cid:48) ) = r · φ ( λ ) . Let ν i ( λ ) = (cid:81) j (cid:54) = i λ ( e ij ) (cid:81) j,k (cid:54) = i λ ( e jk ) and let µ ( v i ) = r ν i ( λ (cid:48) ) /ν i ( λ ) . Then λ (cid:48) ( e ij ) = µ ( v i ) λ ( e ij ) µ ( v j ) . Therefore, φ ∗ is injective. The differentiability of φ ∗ and ( φ ∗ ) − follows from the definition of φ. As a consequence of Theorem 2.2, Lemma 5.1, Lemma 5.6 and Lemma 6.2, we have
Corollary 6.3.
Let T be a tetrahedral triangulation of Σ , with the set of ideal triangles T. Then M T (Σ , ) ∼ = (cid:97) (cid:15) ∈{± } T ∆( T , (cid:15) ) , where each ∆( T , (cid:15) ) is a subset of ∆ = { ( a, b, c ) ∈ R > | a + b + c = 1 } defined as follows.(1) For i ∈ { , . . . , } , let (cid:15) i ∈ {± } T be given by (cid:15) i ( t i ) = − and (cid:15) i ( t j ) = 1 for j (cid:54) = i, and let (cid:15) − i ∈ {± } T be given by (cid:15) − i ( t i ) = 1 and (cid:15) − i ( t j ) = − for j (cid:54) = i. Then ∆( T , (cid:15) i ) = ∆( T , (cid:15) − i ) = { ( a, b, c ) ∈ ∆ | a (cid:54) = b + c, b (cid:54) = a + c and c (cid:54) = a + b } . (2) For { i, j, k, l } = { , . . . , } , let (cid:15) ij ∈ {± } T be given by (cid:15) ij ( t i ) = (cid:15) ij ( t j ) = − and (cid:15) ij ( t k ) = (cid:15) ij ( t l ) = 1 . Then ∆( T , (cid:15) ) = ∆( T , (cid:15) ) = { ( a, b, c ) ∈ ∆ | a (cid:54) = b + c } , ∆( T , (cid:15) ) = ∆( T , (cid:15) ) = { ( a, b, c ) ∈ ∆ | b (cid:54) = a + c } and ∆( T , (cid:15) ) = ∆( T , (cid:15) ) = { ( a, b, c ) ∈ ∆ | c (cid:54) = a + b } . Proof of Theorem 6.1.
Suppose s is a quadruple of positive and negative signs and k ∈ { , , − } . Let T be atetrahedral triangulation of Σ , . Since M T (Σ , ) is dense and open in M (Σ , ) , M sk (Σ , ) (cid:54) = ∅ if and only if M sk (Σ , ) ∩ M T (Σ , ) (cid:54) = ∅ . For (1), by Lemma 5.6, the only possibility for M T (Σ , ) ∩ M s (Σ , ) (cid:54) = ∅ is that s = s ij for some { i, j } ⊂ { , . . . , } . For (2), by Lemma 5.1, the only possibility for M T (Σ , ) ∩ M s (Σ , ) (cid:54) = ∅
19s that either s = s + , in which case λ ( x ) , λ ( y ) and λ ( z ) satisfy the triangular inequality, or s = s i for some i ∈ { , . . . , } , in which case λ ( x ) , λ ( y ) and λ ( z ) satisfy one of the inequalities in (5.2). The proof of (3) isparallel to that of (2).For (4), by symmetry, it suffices to prove the connectedness of M s (Σ , ) , M s (Σ , ) and M s + (Σ , ) . Weconsider the following subsets of ∆ . Let ∆ x ( T , (cid:15) ) (resp. ∆ y ( T , (cid:15) ) and ∆ z ( T , (cid:15) ) ) be the set of points ( a, b, c ) ∈ ∆ such that a > b + c (resp. b > a + c and c > a + b ), let ∆ cx ( T , (cid:15) ) (resp. ∆ cy ( T , (cid:15) ) and ∆ cz ( T , (cid:15) ) ) be the set of points ( a, b, c ) ∈ ∆ such that a < b + c (resp. b < a + c and c < a + b ) and let ∆ c ( T , (cid:15) ) = ∆ cx ( T , (cid:15) ) ∩ ∆ cy ( T , (cid:15) ) ∩ ∆ cz ( T , (cid:15) ) . See Figure 9. By Theorem 2.2, Lemma 5.1, Lemma 5.6 and Corollary 6.3, we have via the lengths coordinate that (cid:1) x (cid:1) x (cid:1) z (cid:1) y (cid:1) xc (cid:1) c Figure 9(1) ∆ x ( T , (cid:15) ) (cid:96) ∆ cx ( T , (cid:15) ) is diffeomorphic to a dense and open subset of M s (Σ , ) , (2) ∆ x ( T , (cid:15) ) (cid:96) ∆ y ( T , (cid:15) ) (cid:96) ∆ z ( T , (cid:15) ) is diffeomorphic to a dense and open subset of M s (Σ , ) , and(3) (cid:96) i =1 ∆ c ( T , (cid:15) i ) is diffeomorphic to a dense and open subset of M s + (Σ , ) . In the rest of the proof, we let T (cid:48) be the tetrahedral triangulation of Σ , obtained from T by doing a simultaneousdiagonal switch S z , let T (cid:48) be the set of ideal triangles of T (cid:48) , and let (cid:15) (cid:48) i and (cid:15) (cid:48) ij ∈ {± } T (cid:48) be sign assignmentsdefined in the same way as (cid:15) i and (cid:15) ij . For ( a, b, c ) ∈ R > , we let [ a, b, c ] . = ( aa + b + c , ba + b + c , ca + b + c ) ∈ ∆ . For the connectedness of M s (Σ , ) , since both ∆ x ( T , (cid:15) ) and ∆ cx ( T , (cid:15) ) are connected, it suffices tochoose two points p and q respectively in ∆ x ( T , (cid:15) ) and ∆ cx ( T , (cid:15) ) and find a path in M s (Σ , ) connecting p and q. Now let p = ( a, b, c ) ∈ ∆ x ( T , (cid:15) ) and let q = ( a (cid:48) , b (cid:48) , c (cid:48) ) ∈ ∆ cx ( T , (cid:15) ) with a (cid:48) > b (cid:48) . By Proposition2.3 and Lemma 5.8, p corresponds to the point p (cid:48) = [ a, b, ( a − b ) /c ] ∈ ∆ x ( T (cid:48) , (cid:15) (cid:48) ) and q corresponds to thepoint q (cid:48) = [ a (cid:48) , b (cid:48) , ( a (cid:48) − b (cid:48) ) /c (cid:48) ] ∈ ∆ x ( T (cid:48) , (cid:15) (cid:48) ) . Since ∆ x ( T (cid:48) , (cid:15) (cid:48) ) is connected, there is a path in ∆ x ( T (cid:48) , (cid:15) (cid:48) ) connecting p (cid:48) and q (cid:48) , giving a path in M s (Σ , ) connecting p and q. For the connectedness of M s (Σ , ) , we let p = ( a, b, c ) ∈ ∆ x ( T , (cid:15) ) and let q = ( a (cid:48) , b (cid:48) , c (cid:48) ) ∈ ∆ y ( T , (cid:15) ) . ByProposition 2.3, Lemma 5.3 and Lemma 5.4, p corresponds to the point p (cid:48) = [ a, b, | a − b | /c ] ∈ ∆ z ( T (cid:48) , (cid:15) (cid:48) ) and q corresponds to the point q (cid:48) = [ a (cid:48) , b (cid:48) , | a (cid:48) − b (cid:48) | /c (cid:48) ] ∈ ∆ z ( T (cid:48) , (cid:15) (cid:48) ) . Since ∆ z ( T (cid:48) , (cid:15) (cid:48) ) is connected, there is a pathin ∆ z ( T (cid:48) , (cid:15) (cid:48) ) connecting p (cid:48) and q (cid:48) , giving a path in M s (Σ , ) connecting p and q. Similarly, any pair of points q ∈ ∆ y ( T , (cid:15) ) and r ∈ ∆ z ( T , (cid:15) ) and any pair of points p ∈ ∆ y ( T , (cid:15) ) and r ∈ ∆ z ( T , (cid:15) ) can respectively beconnected by paths in M s (Σ , ) . Therefore, M s (Σ , ) is connected.Finally, for the connectedness of M s + (Σ , ) , we let p = ( a, b, c ) ∈ ∆ c ( T , (cid:15) ) with a > b and let q =( a (cid:48) , b (cid:48) , c (cid:48) ) ∈ ∆ c ( T , (cid:15) ) with b (cid:48) > a (cid:48) . By Proposition 2.3, Lemma 5.3 and Lemma 5.4, p corresponds to the point p (cid:48) = [ a, b, | a − b | /c ] ∈ ∆ c ( T (cid:48) , (cid:15) (cid:48) ) and q corresponds to the point q (cid:48) = [ a (cid:48) , b (cid:48) , | a (cid:48) − b (cid:48) | /c (cid:48) ] ∈ ∆ c ( T (cid:48) , (cid:15) (cid:48) ) . Since ∆ c ( T (cid:48) , (cid:15) (cid:48) ) is connected, there is a path in ∆ c ( T (cid:48) , (cid:15) (cid:48) ) connecting p (cid:48) and q (cid:48) , giving a path in M s + (Σ , ) connecting p and q. Similarly, all the other pieces can be connected by paths in M s + (Σ , ) , and M s + (Σ , ) isconnected. M od (Σ , ) -action The goal of this section is to prove the ergodicity of the
M od (Σ , ) -action on the non-external connected com-ponents of M (Σ , ) . To use the techniques we used in the previous sections, we need to understand the measureon M (Σ , ) induced by the Goldman symplectic -form in terms the quantities λ ( x ) , λ ( y ) and λ ( z ) . Let T be20 tetrahedral triangulation of Σ , , and let T be the set of ideal triangles of T . For each (cid:15) ∈ {± } T , let ∆( T , (cid:15) ) be the subset of R > defined in Corollary 6.3. Then by Equation (2.2), Lemma 6.2, Corollary 6.3 and a directcalculation, we have the following Proposition 7.1.
For each (cid:15) ∈ {± } T , the -form ω = dλ ( x ) ∧ dλ ( y ) λ ( x ) λ ( y ) + dλ ( y ) ∧ dλ ( z ) λ ( y ) λ ( z ) + dλ ( z ) ∧ dλ ( x ) λ ( z ) λ ( x ) on ∆( T , (cid:15) ) corresponds to the Goldman symplectic -form ω W P on M (Σ , ) , and the measure induced by ω is inthe same measure class of the Lebesgue measure on ∆( T , (cid:15) ) . As a consequence, we have
Proposition 7.2.
For k ∈ {− , , } , the set Ω k (Σ , ) consisting of conjugacy classes of type-preserving represen-tations ρ with the relative Euler class e ( ρ ) = k such that all the tetrahedra triangulation of Σ , are ρ -admissibleis a full measure subset of M k (Σ , ) , and is invariant under the M od (Σ , ) -action.Proof. By the proof of Proposition 5.5, M (Σ , ) \ Ω (Σ , ) is a countable union of Lebesgue measure zerosubsets, hence is of Lebesgue measure zero. Then by Proposition 7.1, M (Σ , ) \ Ω (Σ , ) is a null set in themeasure induced by the Goldman symplectic -form. By the similar argument, Ω (Σ , ) and Ω − (Σ , ) arerespectively full measure subsets of M (Σ , ) and M − (Σ , ) . By Proposition 4.1, since all the simultaneousdiagonal switches act on each Ω k (Σ , ) , so does M od (Σ , ) . Remark . Since Ω (Σ , ) is dense in M (Σ , ) and a representation in Ω (Σ , ) ∩ (cid:96) i =1 M s i (Σ , ) sendseach simple closed curve to a hyperbolic element, by continuity, every representation in (cid:96) i =1 M s i (Σ , ) sendseach simple closed curve to either a hyperbolic or a parabolic element. In [5], Delgado explicitly constructeda family { ρ t } of representations in M (Σ , ) that send every simple closed curve to either a hyperbolic or aparabolic element, and for each ρ t , at least one simple closed curve is sent to a parabolic element. Therefore, therepresentations { ρ t } are in the measure zero subset M (Σ , ) \ Ω (Σ , ) . Let V = { v , . . . , v } be the set of punctures of Σ , . For k ∈ {− , , } and s ∈ {± } V , let Ω sk (Σ , ) =Ω k (Σ , ) ∩ M sk (Σ , ) . By Theorem 6.1 and Proposition 7.2, Theorem 1.4 follows from the following
Theorem 7.4. (1) The
M od (Σ , ) -action on Ω s ij (Σ , ) is ergodic for each { i, j } ⊂ { , . . . , } . (2) The M od (Σ , ) -action on Ω s + (Σ , ) and Ω s − − (Σ , ) is ergodic.(3) The M od (Σ , ) -action on Ω s i (Σ , ) and Ω s − i − (Σ , ) is ergodic for each i ∈ { , . . . , } . Remark . The ergodicity of the
M od (Σ , ) -action on the components of M ± (Σ , ) was first known to Maloni-Palesi-Tan in [22] using the Markoff triple technique. By symmetry, it suffices to prove the ergodicity of the
M od (Σ , ) -action on Ω s (Σ , ) . Let ∆ = { ( a, b, c ) ∈ R > | a + b + c = 1 } , and let ∆ x = { ( a, b, c ) ∈ ∆ | a (cid:54) = b + c } . By Theorem 2.2, Lemma 6.2 and Corollary 6.3,given a tetrahedral triangulation of Σ , , ∆ x is diffeomorphic to a dense and open subset of M s (Σ , ) , wherethe diffeomorphism is given by the quantities λ ( x ) , λ ( y ) and λ ( z ) . Consider the embedding of i : ∆ → R > defined by i (( a, b, c )) = (1 , b/a, c/a ) . Then i (∆ x ) = { (1 , b, c ) ∈ R > | b + c (cid:54) = 1 } . Let Ω X be the subset of i (∆ x ) consisting of the elements coming from Ω s (Σ , ) . As an immediate consequencesof Lemma 5.8, the simultaneous diagonal switches S y and S z act on Ω X by S y (cid:0) (1 , b, c ) (cid:1) = (cid:16) , (1 − c ) b , c (cid:17) and S z (cid:0) (1 , b, c ) (cid:1) = (cid:16) , b, (1 − b ) c (cid:17) . emma 7.6. Let (cid:104) D X (cid:105) be the cyclic subgroup of M od (Σ , ) generated by the Dehn twist D X along the distin-guished simple closed curve X. Then for every k ∈ ( − , , the ellipse E k = { (1 , b, c ) ∈ Ω X | ( b + c − = ( k + 2) bc } is invariant under the action of (cid:104) D X (cid:105) , and for almost every k ∈ ( − , , the action of (cid:104) D X (cid:105) on E k is ergodic.Proof. A direct calculation shows that E k is invariant under the actions of S y and S z for all k ∈ ( − , . Recall that D X = S z S y . Therefore, the ellipse E k is invariant under the (cid:104) D X (cid:105) -action. By lemma 5.8, the action of S y and S z respectively move a point p on E k vertically and horizontally. As show in Figure 10, the an affine transformationof R sending the ellipse E k to a circle C k sends the vertical and the horizontal lines in R respectively to twofamily of parallel lines in C k . As a consequence, for each point p on C k , the angle ∠ pp (cid:48) D X ( p ) is a constant θ k / depending only on k, and the center angle ∠ pOD X ( p ) = 2 ∠ pp (cid:48) D X ( p ) = θ k . Therefore, D X acts on C k by arotation of angle θ k . Since θ k is an irrational multiple of π for almost every k ∈ ( − , , the action of (cid:104) D X (cid:105) isergodic. z S y S bc 𝐷 X bc z S y S k E k C θ k O( ) 𝐷 X ( ) pp p pp’ 𝛺 X Figure 10
Lemma 7.7.
Let D Y D Z be the self-diffeomorphism of Σ , given by the Dehn twist D Z along the distinguishedcurve Z in T followed by the Dehn twist D Y along the distinguished curve Y in D Z ( T ) , and let (cid:104) D Y D Z (cid:105) be thecyclic subgroup of M od (Σ , ) generated by D Y D Z . Then for every k ∈ ( − , , the quartic curve Q k = { (1 , b, c ) ∈ Ω X | ( b + c ) ( b + c − = ( k + 2) bc } is invariant under the action of (cid:104) D Y D Z (cid:105) , and for almost every k ∈ ( − , , the action of (cid:104) D Y D Z (cid:105) on Q k isergodic.Proof. A direct calculation shows that Q k is the S x -image of E k . Since D Y D Z = S x S z S y S x = S x D X S x , themap S z : E k → Q k is Z -equivariant, where ∈ Z acts on E k by D X and acts on Q k by D Y D Z . By Lemma 7.6, (cid:104) D Y D Z (cid:105) acts on Q k for every k ∈ ( − , , and the action is ergodic for almost every k ∈ ( − , . Proof of Theorem 7.4 (1).
We show that every
M od (Σ , ) -invariant measurable function F : Ω X → R is almosteverywhere a constant. Consider the following region R = { (1 , b, c ) ∈ Ω X | ( b + c − < bc, b + c < } in Ω X inclosed by the parabola P = { (1 , b, c ) ∈ Ω X | ( b + c − = 4 bc } and the line segment L = { (1 , b, c ) ∈ Ω X | b + c = 1 } . We claim that each point p = (1 , b , c ) in R is an intersection an ellipse E k and a quantic curve Q k for some k , k ∈ ( − , . Indeed, we can let k = ( b + c − b c − and let k = ( b + c ) ( b + c − b c − . Since ( b + c − < b c , k ∈ ( − , , and since b + c < , k ∈ ( − , . A direct calculation showsthat the intersection of E k and Q k is transverse at p, i.e., the gradients ∇ E k ( p ) and ∇ Q k ( p ) span the tangent22pace of Ω X at p. Then by Lemma 7.6 and Lemma 7.7, the restriction of F to R is almost everywhere a constant.For p ∈ Ω X , let O ( p ) be the M od (Σ , ) -orbit of p. To show that the F is almost everywhere a constant in Ω X , itsuffices to show that O ( p ) ∩ R (cid:54) = ∅ for almost every p in Ω X . Let R (cid:48) be the region in Ω X enclosed by parabola P, i.e., R (cid:48) = { (1 , b, c ) ∈ Ω X | ( b + c − < bc } . Then R (cid:48) is foliated by the ellipses { E k } . We note that theparabola P is the i -image of the inscribe circle C of ∆ . Then by the Trace Reduction Algorithm, Lemma 5.9 andProposition 7.2, for almost every p in Ω X , there is a composition φ of finitely many, say m, simultaneous diagonalswitches such that φ ( p ) ∈ R (cid:48) . By Proposition 4.1, if m is even, then φ ∈ M od (Σ , ) and O ( p ) ∩ R (cid:48) (cid:54) = ∅ ; andif m is odd, then φ (cid:48) = S y φ ∈ M od (Σ , ) . Since S y keeps invariant an ellipse E k ⊂ R (cid:48) passing through φ ( p ) ,φ (cid:48) ( p ) = S y φ ( p ) ∈ E k ⊂ R (cid:48) , and hence O ( p ) ∩ R (cid:48) (cid:54) = ∅ . Finally, by Lemma 7.6, for almost every p in R (cid:48) , there is n such that D nX ( p ) ∈ E k ∩ R ⊂ R. By symmetry, it suffices to prove the ergodicity of the
M od (Σ , ) -action on Ω s + . The strategy is to find twotransversely intersecting families of curves { E X,k } and { E Y,k } foliating M s + (Σ , ) such that the (cid:104) D X (cid:105) -actionon almost every E X,k and the (cid:104) D Y (cid:105) -action on almost every E Y,k is ergodic. Let T be an tetrahedral triangulationof Σ , , and let T be the set of ideal triangles of T . Let ∆ = { ( a, b, c ) ∈ R > | a + b + c = 1 } , and for (cid:15) ∈ {± } T , let ∆ c ( T , (cid:15) ) = { ( a, b, c ) ∈ ∆ | a < b + c, b < a + c, c < b + a } . By Theorem 2.2, Lemma 5.1 and Corollary 6.3, (cid:96) i =1 ∆ c ( T , (cid:15) i ) is diffeomorphic to a dense and open subset of M s + (Σ , ) , where the diffeomorphism is givenby the quantities λ ( x ) , λ ( y ) and λ ( z ) . We define the embedding i X : (cid:96) i =1 ∆ c ( T , (cid:15) i ) → R by i X (( a, b, c )) = (1 , b/a, c/a ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )(1 , − b/a, − c/a ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )(1 , − b/a, c/a ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )(1 , b/a, − c/a ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) ) . For i ∈ { , . . . , } , we let Ω X,i be the subset of i X (∆ c ( T , (cid:15) i )) consisting of the elements coming from Ω s + (Σ , ) , and let Ω X = (cid:96) i =1 Ω X,i . (See Figure 11.) bc X,1
X,k E 𝛺 X,3X,2 X,4 𝐷 X ( ) pp 𝛺𝛺 𝛺
Figure 11
Lemma 7.8.
For every k ∈ ( − , , the ellipse E X,k = { (1 , b, c ) ∈ Ω X | b + c − kbc } is invariant under the action of (cid:104) D X (cid:105) , and for almost every k ∈ ( − , , the action of (cid:104) D X (cid:105) on E X,k is ergodic. roof. For ( x, y, z ) ∈ R , let | ( a, b, c ) | = ( | a | , | b | , | c | ) . By Lemma 5.3, the action of S y and S z on Ω X satisfies (cid:12)(cid:12) S y (cid:0) (1 , b, c ) (cid:1)(cid:12)(cid:12) = (cid:16) , (cid:12)(cid:12)(cid:12) c − b (cid:12)(cid:12)(cid:12) , | c | (cid:17) and (cid:12)(cid:12) S z (cid:0) (1 , b, c ) (cid:1)(cid:12)(cid:12) = (cid:16) , | b | , (cid:12)(cid:12)(cid:12) b − c (cid:12)(cid:12)(cid:12)(cid:17) . Therefore, we have (cid:12)(cid:12) D X (cid:0) (1 , b, c ) (cid:1)(cid:12)(cid:12) = (cid:16) , (cid:12)(cid:12)(cid:12) c − b (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ( c − b (cid:1) − c (cid:12)(cid:12)(cid:12)(cid:17) . We claim that D X (cid:0) (1 , b, c ) (cid:1) = (cid:16) , c − b , ( c − b (cid:1) − c (cid:17) . (7.1)If (7.1) is true, then a direct calculation shows that D X ((1 , b, c )) is on E X,k . To verify (7.1), we let T (cid:48) be the tetrahedral triangulation obtained from T by doing S y , and let T (cid:48)(cid:48) be thetetrahedral triangulation obtained from T (cid:48) by doing S z . Let (cid:15) (cid:48) and (cid:15) (cid:48)(cid:48) respectively be the signs of ρ assigned tothe ideal triangles of T (cid:48) and T (cid:48)(cid:48) . In T , T (cid:48) and T (cid:48)(cid:48) , we denote uniformly by t i the ideal triangle disjoint from thepuncture v i . If p = (1 , b, c ) ∈ Ω X, , i.e. i − X ( p ) ∈ ∆ c ( T , (cid:15) ) , then we consider the following cases.Case 1: c > and c − b > . In this case, since (cid:15) ( t ) = − and c > , we have by Proposition 2.3 that (cid:15) (cid:48) ( t ) = − . Since c − b > , by Proposition 2.3 again, (cid:15) (cid:48)(cid:48) ( t ) = − . Therefore, D X ( i − X ( p )) ∈ ∆ c ( T (cid:48)(cid:48) , (cid:15) ) , and(7.1) follows.Case 2: c > and c − b < . In this case, by Proposition 2.3, (cid:15) (cid:48) ( t ) = − and (cid:15) (cid:48)(cid:48) ( t ) = − . Therefore, D X ( i − X ( p )) ∈ ∆ c ( T (cid:48)(cid:48) , (cid:15) ) , and and (7.1) follows.Case 3: c < and c − b > . In this case, by Proposition 2.3, (cid:15) (cid:48) ( t ) = − and (cid:15) (cid:48)(cid:48) ( t ) = − . Therefore, D X ( i − X ( p )) ∈ ∆ c ( T (cid:48)(cid:48) , (cid:15) ) , and and (7.1) follows.Case 4: c < and c − b < . In this case, by Proposition 2.3, (cid:15) (cid:48) ( t ) = − and (cid:15) (cid:48)(cid:48) ( t ) = − . Therefore, D X ( i − X ( p )) ∈ ∆ c ( T (cid:48)(cid:48) , (cid:15) ) , and and (7.1) follows.The verification of (7.1) for p in Ω X, , Ω X, and Ω X, is similar, and is left to the readers.By (7.1), the action of D X on E X,k is a horizontal translation followed by a vertical translation. See Figure11. By doing a suitable affine transform, the (cid:104) D X (cid:105) -action is a rotation of an angle θ k on a circle, where θ k is anirrational multiple of π for almost every k. Therefore, for almost every k ∈ ( − , , the (cid:104) D X (cid:105) -action on E X,k isergodic.Consider the embedding i Y : (cid:96) i =1 ∆ c ( T , (cid:15) i ) → R by i Y (( a, b, c )) = ( a/b, , c/b ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )( − a/b, , c/b ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )( − a/b, , − c/b ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) )( a/b, , − c/b ) , if ( a, b, c ) ∈ ∆ c ( T , (cid:15) ) For i ∈ { , . . . , } , we let Ω Y,i be the subset of i Y (∆ c ( T , (cid:15) i )) consisting of the elements coming from Ω s +1 (Σ , ) , and let Ω Y = (cid:96) i =1 Ω Y,i . Then we have the following lemma whose proof is similar to that of Lemma 7.8.
Lemma 7.9.
For every k ∈ ( − , , the ellipse E Y,k = { ( a, , c ) ∈ Ω Y | a + c − kac } is invariant under the action of (cid:104) D Y (cid:105) , and for almost every k ∈ ( − , , the action of (cid:104) D Y (cid:105) on E Y,k is ergodic.Proof of Theorem 7.4 (2).
A direct calculation shows that the two family of curves { i − X ( E X,k ) } and { i − Y ( E Y,k ) } transversely intersect. Then by Lemma 7.8 and Lemma 7.9, the M od (Σ , ) -action on M s + (Σ , ) is ergodic.24 .3 A proof of Theorem 7.4 (3) Proof.
By symmetry, it suffices to prove the ergodicity of the
M od (Σ , ) -action on Ω s (Σ , ) . We let ∆ x = { ( a, b, c ) ∈ ∆ | a > b + c } , ∆ y = { ( a, b, c ) ∈ ∆ | b > a + c } and ∆ z = { ( a, b, c ) ∈ ∆ | c > a + b } . By Theorem2.2, Lemma 6.2 and Corollary 6.3, given a tetrahedral triangulation of Σ , , ∆ x (cid:96) ∆ y (cid:96) ∆ z is diffeomorphic to adense and open subset of M s (Σ , ) , where the diffeomorphism is given by the quantities λ ( x ) , λ ( y ) and λ ( z ) . Let R = { ( s, t ) ∈ R | s (cid:54) = 0 , t (cid:54) = 0 , s + t (cid:54) = 0 } , and consider the two-fold covering map ψ : R → ∆ x (cid:96) ∆ y (cid:96) ∆ z defined by ψ (( s, t )) = (cid:2) sinh | s | , sinh | t | , sinh | s + t | (cid:3) , where [ a, b, c ] . = ( aa + b + c , ba + b + c , ca + b + c ) . Let Ω be the subset of ∆ x (cid:96) ∆ y (cid:96) ∆ z consisting of the elements com-ing from Ω s (Σ ,t ) , and let Ω (cid:48) = ψ − (Ω) . Then by Proposition 7.2, Ω is invariant under the M od (Σ , ) -action.Recall that M od (Σ , ) is isomorphic to a free group F of rank two generated by the Dehn twists D X and D Y . (See e.g. [7].) It is well known that F is isomorphic to the quotient group Γ(2) / ± I, where Γ(2) = (cid:110) A ∈ SL (2 , Z ) (cid:12)(cid:12)(cid:12) A ≡ (cid:18) (cid:19) ( mod (cid:111) is the mod- congruence subgroup of SL (2 , Z ) , and the matrices (cid:20) (cid:21) and (cid:20) (cid:21) correspond to thegenerators of F . This induces a group homomorphism π : Γ(2) → M od (Σ , ) defined by π (cid:16) (cid:20) (cid:21) (cid:17) = D X and π (cid:16) (cid:20) (cid:21) (cid:17) = D Y . By Lemma 5.3 and a direct calculation, ψ : Ω (cid:48) → Ω is π -equivariant, i.e. ψ ◦ A = π ( A ) ◦ ψ for all A ∈ Γ(2) , where the Γ(2) -action on Ω (cid:48) is the standard linear action. By Moore’s Ergodicity Theorem [21], the Γ(2) -actionon R , hence on Ω (cid:48) , is ergodic. Therefore, the M od (Σ , ) -action on Ω is ergodic. A Equivalence of extremal and Fuchsian representations
Proposition A.1 below is a consequence of Goldman ([11], Theorem D) and is stated without proof in [4]. Thepurpose of this appendix is to include a proof of it for the readers’ interest, where the argument is from a discussionwith F. Palesi and M. Wolff.
Proposition A.1.
A type-preserving representation ρ : π (Σ g,n ) → P SL (2 , R ) is extremal, i.e., | e ( ρ ) | = 2 g − n, if and only if ρ is Fuchsian.Proof. By Goldman [11], Theorem D, a representation ρ is maximum if and only if ρ is Fuchsian and the quotient H /ρ ( π (Σ g,n )) is homeomorphic to Σ g,n . Therefore, to prove the Proposition, it suffices to rule out the possibilitythat ρ is non-maximum, Fuchsian and H /ρ ( π (Σ g,n )) = Σ g (cid:48) ,n (cid:48) (cid:29) Σ g,,n, , which we will do using a contradiction.Now since H /ρ ( π (Σ g,n )) = Σ g (cid:48) ,n (cid:48) , there is an isomorphism φ : π (Σ g (cid:48) ,n (cid:48) ) → ρ ( π (Σ g,n )); and since ρ is type-preserving, φ ( π (Σ g (cid:48) ,n (cid:48) )) = ρ ( π (Σ g,n )) contains n parabolic elements from the primitiveperipheral elements of π (Σ g,n ) . On the other hand, since the only possible parabolic elements of a Fuchsiansubgroup of
P SL (2 , R ) are from the peripheral elements, the composition φ − ◦ ρ : π (Σ g,n ) → π (Σ g (cid:48) ,n (cid:48) ) must send the primitive peripheral elements of π (Σ g,n ) to the primitive peripheral elements of π (Σ g (cid:48) ,n (cid:48) ) . Thisis impossible when n > n (cid:48) , since π (Σ g (cid:48) ,n (cid:48) ) has only n (cid:48) primitive peripheral elements. For the case n < n (cid:48) , werecall the fact that in the first homology H (Σ , R ) of a punctured surface Σ , the full set of vectors represented by25he primitive peripheral elements of π (Σ) are linearly dependent, but the vectors in any proper subset of it arelinearly independent. Therefore, the induced isomorphism ( φ − ◦ ρ ) ∗ : H (Σ g,n ; R ) → H (Σ g (cid:48) ,n (cid:48) ; R ) sends a set of linearly dependent vectors represented by the primitive peripheral elements of π (Σ g,n ) to a set oflinearly indecent vectors, which is a contradiction. B Relationship with Goldman’s work on one-punctured torus
In this appendix, we show that the results concerning representations of relative Euler class ± in this paper canalso be seen, and more straightforwardly, as consequences of some previous results of Goldman ([13], Chapter 4),where the argument presented here is due to the anonymous referee.In [13], Goldman considers SL (2 , R ) -representations of the one-puncture torus group π (Σ , ) , which is thefree group of two generators (cid:104) X, Y (cid:105) . The character space M red (Σ , ) of reducible representations ρ : π (Σ , ) → SL (2 , R ) satisfy tr ( ρ [ X, Y ]) = 2 , and hence could be described by the equation x + y + z − xyz − (B.1)where x = tr ( ρ ( X )) , y = tr ( ρ ( Y )) and z = tr ( ρ ( XY )) . On the other hand, the fundamental group of thefour puncture sphere π (Σ , ) ∼ = (cid:104) A, B, C, D | ABCD (cid:105) , where the generators are the four primitive periph-eral elements corresponding to the four punctures. If ρ : π (Σ , ) → P SL (2 , R ) is type-preserving, then | tr ( ρ ( A )) | = | tr ( ρ ( B )) | = | tr ( ρ ( C )) | = | tr ( ρ ( D )) | = 2; and if e ( ρ ) = ± , then one can lift ρ to a repre-sentation (cid:101) ρ : π (Σ , ) → SL (2 , R ) such that tr ( (cid:101) ρ ( A )) tr ( (cid:101) ρ ( B )) tr ( (cid:101) ρ ( C )) tr ( (cid:101) ρ ( D )) < . Hence the characterspaces M ± (Σ , ) can be described by the equation x + y + z + xyz − (B.2)where x = tr ( ρ ( AB )) , y = tr ( ρ ( BC )) and z = tr ( ρ ( CA )) . (See e.g. [12, 22] for more details.) Comparing(B.1) and (B.2), it is clear that M red (Σ , ) ∼ = M ± (Σ , ) . Moreover, the mapping class group actions are commensurable and the variables x, y, z correspond in each case tothe traces of simple closed curves on the surface, hence all the results known for M red (Σ , ) can be translated tothe results on M ± (Σ , ) . To be more precise, by ([13], Chapter 4), M red (Σ , ) has five connected components, one of which is compactcorresponding to M s ± (Σ , ) and four of which are non-compact corresponding to M s i ± (Σ , ) . A full measuresubset of the characters in the non-compact components have all coordinates x, y, z strictly greater than inabsolute value. Each coordinate correspond to the trace of the image of a simple closed curve. Starting from arepresentation in one of these components and using the transitivity of the mapping class group action on the setof simple closed curves, one gets that every simple closed curve is sent to an hyperbolic element. Therefore, a fullmeasure subset of representations in the non-compact components are counterexamples to Bowditch’s question.Finally, the ergodicity of the P SL (2 , Z ) -action on the non-compact components is already proved in ([13], Chapter4), implying the M od (Σ , ) -action on M s i ± (Σ , ) . References [1] Bonahon, F.,
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