The intrinsic asymmetry and inhomogeneity of Teichmuller space
aa r X i v : . [ m a t h . G T ] J a n The intrinsic asymmetry and inhomogeneityof Teichm¨uller space
Benson Farb and Shmuel Weinberger ∗ October 22, 2018
Let S = S g,n be a connected, orientable surface of genus g ≥ n ≥ S ) denote the corresponding Teichm¨uller space, and let Mod( S ) denote the mappingclass group of S . Understanding the analogy of Teich( S ) with symmetric spaces is a well-known theme. Recall that a complete Riemannian manifold X is symmetric if it is symmetricat each point x ∈ X : the map γ ( t ) γ ( − t ) which flips geodesics about x is an isometry.Symmetric spaces X are homogeneous : the isometry group Isom( X ) acts transitively on X . In his famous paper [Ro], Royden studied the possible symmetry and homogeneity ofTeich( S ), endowed with the Teichm¨uller metric. Theorem 1.1 (Royden [Ro]) . Suppose S is closed of genus at least , and let Teich( S ) beTeichm¨uller space endowed with the Teichm¨uller metric d T eich . Thena. (Teich( S ) , d Teich ) is not symmetric at any point.b. Isom(Teich( S ) , d Teich ) contains Mod( S ) (modulo its center if S is closed of genus )as a subgroup of index . Note that the conclusion of Theorem 1.1 is false when genus( S ) = 1, as d Teich in thiscase is the hyperbolic metric on the upper half-plane. Earle-Kra [EK] extended part (b) ofRoyden’s Theorem to arbitrary surfaces of finite type. Royden deduced Theorem 1.1 froma detailed analysis of the fine structure of the space QD ( M ) of unit norm holomorphicquadratic differentials on a Riemann surface M . In particular, he found an embedding of M in QD ( M ) and characterized it by the degree of Holder regularity of the norm on QD ( M )at the points of the embedding.The Teichm¨uller metric is a complete Finsler metric, under which moduli space M ( S ) :=Teich( S ) / Mod( S ) has finite volume (see the proof of Theorem 8.1 of [Mc]) . There aremany other complete, finite volume, Mod( S )-invariant Finsler (indeed Riemannian) metricson Teich( S ), each with special properties. Examples include the Kahler-Einstein metric, ∗ Both authors are supported in part by the NSF. A Finsler metric determines a unique volume form by declaring the unit Finsler ball at each point tohave volume 1.
Theorem 1.2.
Let S = S g,n be a surface, and let d be any complete, finite covolume, Mod( S ) -invariant Finsler (e.g. Riemannian) metric on Teich( S ) . Thena. If g ≥ then (Teich( S ) , d ) is not symmetric at any point.b. If g − n ≥ then Isom(Teich( S ) , d ) contains Mod( S ) (modulo its center if ( g, n ) =(1 , or (2 , ) as a subgroup of finite index. In this way Teichm¨uller space exhibits a kind of intrinsic asymmetry and inhomogeneity.The number 3 g − n plays the role of “ Q -rank” in this context. Theorem 1.2(b) is sharp:the conclusion is false whenever 3 g − n <
2, and indeed the corresponding Teichm¨ullerspaces admit hyperbolic metrics.
Remarks.
1. The proof of Theorem 1.2 gives immediately that the hypothesis can be weakened toallow d to be any Γ-invariant metric, for any finite index subgroup Γ < Mod( S ). Assuch subgroups are typically torsion free, this shows that the phenomenon of inhomo-geneity and asymmetry is not the result of the constraints imposed by having finiteorder symmetries of the metric.2. Ivanov proved (see, e.g., [I2]) that the moduli space of Riemann surfaces Teich( S ) / Mod( S )never admits a locally symmetric metric when S is closed and genus( S ) ≥
2. If it didadmit such a metric, then Teich( S ) would admit a complete, finite covolume, Mod( S )-invariant metric which is symmetric at every point. Thus Theorem 1.2(a) gives a newproof, and generalization, of Ivanov’s theorem.The proof of Theorem 1.2(a) relies on Theorem 1.2(b). One key ingredient in our proofsis Smith theory. This is not the first time Smith theory has been used to analyze actions onTeichm¨uller space: Fenchel used it in the 1940’s to analyze certain periodic mapping classes.We conjecture that the index in Theorem 1.2(b) can be taken to depend only on S .More strongly, one might hope that it can always be taken to be 2. As evidence towardsthis strongest possible conjecture, we can prove it in the “ Q -rank 2” case. Theorem 1.3.
Let S be the twice-punctured torus or the -punctured sphere. Then theindex in Theorem 1.2(b) can be taken to be . Acknowledgements.
We are grateful to Tom Church, Alex Eskin and Dan Margalit fortheir numerous useful and insightful comments. We are especially grateful to Yair Minsky,whose persistence in pinning us down, and whose many comments and corrections, greatlyimproved this paper. 2
Proof of Theorem 1.2(b)
By the Myers-Steenrod Theorem (or, in the general Finsler case, Deng-Hou [DH]), the group I := Isom(Teich( S ) , d ) is a Lie group, possibly with infinitely many components, actingproperly discontinuously on Teich( S ). We remark that Theorem 2.2 of [DH] states that anyisometry of a Finsler metric is necessarily a diffeomorphism; we will use this smoothnesslater.Let I denote the connected component of the identity of I ; note that I is normal in I . Let µ denote the measure on Teich( S ) (and the induced measure on Teich( S ) / Mod( S ))induced by the Finsler metric d . We are assuming that µ (Teich( S ) / Mod( S )) < ∞ .If I is discrete, then we claim that [ I : Mod( S )] < ∞ . To see this, let K I be a measure-theoretic fundamental domain for the I -action on Teich( S ). By this we mean that:1. The complement of S g ∈ I g · K I in Teich( S ) has µ -measure 0, and2. For all nontrivial g ∈ I , we have µ ( g · K I ∩ K I ) = 0.Note that measurable fundamental domains always exist for properly discontinuous ac-tions. Let K Mod( S ) be a measure-theoretic fundamental domain for Mod( S ). Let { g i } be acollection of coset representatives for Mod( S ) in I . Then we can choose K Mod( S ) so that K Mod( S ) = [ i g i · K I . Notice that this union is a disjoint union, up to sets of measure zero, by Condition (2) inthe definition of fundamental domain. Since by assumption µ ( K I ) and µ ( K Mod( S ) ) are bothfinite, it follows that this must be a finite union, i.e. [ I : Mod( S )] < ∞ .So suppose that I is not discrete. Myers-Steenrod then gives that the dimension of I ispositive, and so I is a connected, positive-dimensional Lie group. Let Γ := Mod( S ), andlet Γ := I ∩ Mod( S ). We have the following exact sequences:1 → I → I → I/I → → Γ → Γ → Γ / Γ → Step 1 ( Γ is a lattice in I ): We begin by replacing Γ (hence Γ ) by a torsion-freesubgroup of finite index, but keep the notation Γ (resp. Γ ) for this new group. While notformally necessary, this assumption will make “orbifold technicalities” disappear; the pointis that quotients of Teich( S ) and associated bundles by Γ will be manifolds.We begin by replacing the proper action of I on Teich( S ) by a free action on an associatedspace F (Teich( S )), which we now describe.For any smooth, connected, n -dimensional Finsler manifold M we have the natural unitsphere-bundle over M , which is a sub-bundle of the tangent bundle T M , with fiber over m ∈ M the unit Finsler sphere S m of T M m . We also have the associated bundle E → M whose fiber is the n -fold product S nm . Let F ( M ) denote the sub-bundle of this bundle with3ber the set of n -tuples of distinct points of S m that span T M m . The group Isom( M ) clearlyacts on F ( M ) by homeomorphisms.The exponential map on a Finsler manifold is a local diffeomorphism (see, e.g. [DH],Lemma 1.1). Since we also have that Finsler isometries take geodesic rays to geodesic rays,we see that the set of points of M for which a Finsler isometry is the identity and hasderivative the identity is both open and closed. Thus the action of Isom( M ) on F ( M ) isfree.Now, we will want to apply the Slice Theorem to this action (see below), and to do sowe need it to preserve some smooth structure on F ( M ). To this end, first note that M isassumed to be a smooth manifold, and since Isom( M ) acts properly on M , we have thatIsom( M ) actually preserves some Riemannian metric on M . We then note that there isan Isom( M )-equivariant homeomorphism from F ( M ) to the bundle of unit n -frames over M , and so the action of Isom( M ) on F ( M ) preserves the pullback of the standard smoothstructure on the latter bundle. We remark that the reason we do not simply replace thegiven Finsler metric with this invariant Riemannian metric is that the finiteness of volume of M in the Finsler metric need not imply such finiteness for the invariant Riemannian metric.We now wish to construct an Isom( M )-invariant measure on F ( M ). To this end, notethat the bundle E → M discussed above is locally a product U × S n , where U is a neigh-borhood in M . The Finsler metric on M determines a volume form on M , which inducesa measure ν on U . On S , we have an induced measure µ which is given infinitessimally bythe rule that, for a subset A ⊆ S m , the measure is given by the measure (induced by theFinsler norm of A on T M m ) of the Euclidean cone of A , normalized so that the measure of S m equals 1. The local product measure ν × µ then gives an Isom( M )-invariant measureon E , which in turn induces an Isom( M )-invariant measure on F ( M ). By construction, thepushforward of this measure under the natural projection F ( M ) → M is the measure on M induced by the given Finsler metric; in particular, if M is assumed to have finite measurethen F ( M ) has finite measure.Now let M = Teich( S ). Pick x ∈ F ( M ), and consider the I -orbit O x . The SliceTheorem for proper group actions (see, e.g., [DK], 2.4.1) in this context asserts that there isan I -invariant tubular neighborhood V of O x in F (Teich( S )) that is a homogeneous vectorbundle π : V → O x . The measure on F ( M ) constructed above restricts to a measure on V ,and π pushes this forward to a left-invariant measure on O x , which we can identify with aleft-invariant measure on I . Note that all left-invariant measures on I are proportional, byuniqueness of Haar measure. The key property we will use is that if a subset A ⊆ I hasinfinite measure then π − ( A ) has infinite measure.Choose any fiber D of the bundle V → O x , so that V is the I -orbit of D . Since I is theconnected component of the identity of I , and since I is a closed subgroup of I , we havethat V can be written as a disjoint union of I -orbits of D , one for each element of π ( I ).Note that Γ / Γ ⊆ π ( I ). Thus V /
Γ is given by the image of the I -orbit W of D under theprojection F (Teich( S )) → F (Teich( S )) / Γ = (Teich( S ) / Γ) = F ( M ( S )) . Since the I -action on F (Teich( S )) is free, this projection is a measure-preserving home-omorphism when restricted to W . Now if I / Γ had infinite measure, then so would W (by the discussion above), and thus so would F ( M ( S )). On the other hand the map4 ( M ( S )) → M ( S ) is measure-preserving by the construction of the measure on F ( M ( S ))(see above). This gives that M ( S ) has infinite measure, contradicting the given. We con-clude that I / Γ has finite measure, as desired. Step 2 ( I is semisimple with finite center): For any connected Lie group G there isan exact sequence 1 → G sol → G → G ss → G sol denotes the solvable radical of G (i.e. the maximal connected, normal, solvableLie subgroup of G ), and where G ss is the connected semisimple Lie group G/G sol .We can apply this setup with G = I . We now follow exactly the argument of the proofof Proposition 3.3 of [FW], with the only change being that here Γ is not assumed torsionfree. First, a theorem of Raghunathan gives a unique maximal normal solvable subgroup Γ sol0 of Γ . Since Γ sol0 is unique it is characteristic in Γ , hence normal in Γ. The main theorem of[BLM] states that any solvable subgroup of Γ = Mod( S ) has an abelian subgroup of finiteindex. Hence Γ sol0 has a torsion-free, normal (in Γ) finite-index abelian subgroup, whichwe denote by N . We will prove below that Γ = Mod( S ) has no infinite normal abeliansubgroups, from which we will conclude that N is trivial, so that Γ sol0 is finite.Now the next part of the argument of Proposition 3.3 of [FW] quotes a theorem ofPrasad. In applying Prasad’s theorem, we used the fact that the sum of the ranks of theabelian quotients of the derived series of Γ sol0 equals 0. While in [FW] this followed from thefact that Γ sol0 = 0, we only need that Γ sol0 is finite, which we have just proven. We may thusquote Prasad’s theorem and conclude, as in [FW], that I sol0 is compact, and that the center Z ( I ) is finite.Given this, we finish the proof of Step 2 as follows. Since any compact Lie group is theproduct of a simple Lie group and a torus, I sol0 must be a torus. Further, as is preciselyargued in Proposition 3.3 of [FW], the conjugation action of the connected group I ss0 on thetorus I sol0 , which has discrete automorphism group, must be trivial, so that I sol0 must be adirect factor of I . We thus need to rule this out when I sol0 is positive-dimensional.So if we can prove that Mod( S ) has no infinite, normal abelian subgroup A , and if wecan rule out that I sol0 is a torus direct factor of I , then we have completed Step 2. We beginwith the first claim.By the classification of abelian subgroups of Mod( S ) (see [BLM] or [I1]), any abelian sub-group A , after perhaps being replaced by a finite index characteristic subgroup if necessary,either is cyclic with a pseudo-Anosov generator or there is a unique maximal finite collection C of simple closed curves, called the canonical reduction system of A , left invariant (setwise)by each a ∈ A . In the first case, the normalizer of A is virtually cyclic, a contradiction, sosuppose we are in the latter case. Then for any f ∈ Mod( S ) the canonical reduction systemfor f Af − is f ( C ). The result now follows by picking an f such that f ( C ) = C . Thus A must be trivial.We now rule out that I contains a positive-dimensional torus as a direct factor. Supposeit does. Then Z ( I ) is positive-dimensional. Since Z ( I ) is a positive-dimensional abelianLie group, and we can write Z ( I ) = A × ( S ) d × R k for some d > , k ≥
0. Now let T denote the maximal compact subgroup of the connected component of the identity of Z ( I ).From the above description of Z ( I ), it is clear that T is characteristic in Z ( I ), hence in I . Since I is normal in I , we have that the conjugation action of Γ = Mod( S ) leaves5 invariant. Thus the action of Γ on Teich( S ) leaves Fix( T ) invariant. Since T is actingsmoothly, Fix( T ) is a manifold. Since Teich( S ) is contractible, Fix( T ) is acyclic (see, e.g.,[Br], Theorem 10.3). As T is positive dimensional and connected, and since the T -action onTeich( S ) is faithful, dim(Fix( T )) < dim(Teich( S )) −
1. Note that the action of Mod( S ) onFix( T ) is properly discontinuous, being the restriction of the properly discontinuous actionof Mod( S ) on Teich( S ). Case A ( dim(Fix( T )) > ): We claim that there is a contractible manifold Z of dimensiondim(Fix( T )) + 1 < dim(Teich( S )) on which Mod( S ) acts properly discontinuously. Giventhis, we recall that Despotovic [D] proved that Mod( S ) admits no properly discontinuousaction on any contractible manifold of dimension < dim(Teich( S )), giving us a contradiction.Thus T would be trivial, and so I is semisimple with finite center.We now prove the claim. Note that by Smith theory (see [Br]) Fix( T ) is acyclic. Wewant to replace this Mod( S )-manifold by another one that is contractible, i.e. we want tokill the fundamental group. If Fix( T ) / Mod( S ) is compact, then the kernel that we aretrying to kill is finitely normally generated, and the construction is standard: one kills theelements of this kernel by surgering the circles, giving rise to new homology = homotopy indimension 2, which can then be killed by surgering the 2-spheres as well (see Section 3 of[H]). Taking the product of Fix( T ) with R if dim(Fix( T )) = 4, the 2-spheres needed at thisstage can be embedded by general position since dim(Fix( T ) × R ) > T ) / Mod( S ) is not compact, then one wants to do the same argument, but onehas to be careful to make sure that the circles and 2-spheres that one wants to surger donot accumulate. However, by replacing Fix( T ) by Fix( T ) × R , this problem disappears: onesimply does the i th surgery at the “height” Fix( T ) ×{ i } , producing at height i a 2-dimensionalhomology class. This class can be represented by a sphere which lies in a compact regionthat is above level i − / i of them pass through level i . Surgering thisfree basis for the homology of the previous stage gives us back an acyclic manifold, whichis now, in addition, simply connected, and, thus, contractible. Case B ( dim(Fix( T )) ≤ ): In this case the proper action of Mod( S ) on the acyclic spaceFix( T ) implies that the virtual cohomological dimension vcd(Mod( S )) satisfiesvcd(Mod( S )) ≤ dim(Fix( T )) ≤ . But then by the formulas for vcd(Mod( S )), given for example in Theorem 6.4 of [I2], andsince 3 g − n ≥ g, n ) to be one of { (2 , , (0 , , (0 , , (1 , , (1 , } . As the action of Mod( S ) on Fix( T ) is properly discon-tinuous, this rules out dim(Fix( T )) = 0. Now Fix( T ) has even codimension in Teich( S )(see [Br], Theorem 10.3), and so is even-dimensional. As we are assuming in this case thatdim(Fix( T )) ≤
3, it follows that dim(Fix( T )) = 2. But this would imply that Mod( S ) has aa (closed or open) surface group as a subgroup of finite index. It cannot, however, becausefor instance in these cases Mod( S ) contains both Z , eliminating all surfaces but the torus,and also a rank 2 free group, eliminating the torus.6 tep 3 ( I has no compact factors): Let K be the maximal compact factor of I .Since I is semisimple with finite center, K is characteristic in I . Since I is normal in I ,it follows that K is invariant under conjugation by any element of I . Further, note that K is semisimple since I is semisimple.Since K is compact, we have that K ∩ Γ = K ∩ Γ is a finite normal subgroup of Γ.Replacing Γ by a finite index subgroup, which we will also denote by Γ, we can assumethat K ∩ Γ is trivial. As K is invariant by conjugation by elements of I , we have an exactsequence 1 → K → h K, Γ i → Γ → I generated by K and Γ. As explained inIV.6 of [Bro], any exact sequence 1 → A → B → C → ρ : C → Out( A ) and a cocycle η ∈ H ( C, Z ( A )), where Z ( A ) denotes the center of A , and is a C -module via the action of ρ . In the case (4), we have that both Z ( K ) and Out( K ) are finite since K is semisimple.We may thus pass to a finite index subgroup Λ < Γ so that ρ has trivial image. Now let1 → Z ( K ) → b Λ → Λ → η ∈ H (Λ , Z ( K ))corresponding to the extension (4) restricted to Λ. Note that this exact sequence defines atrivial cocycle. Note also that b Λ ⊆ h K, Λ i . Now Z ( K ) is central both in b Λ and in K . Wethus have a split exact sequence1 → K/Z ( K ) → h K, b Λ i → b Λ → ρ has trivial image, we can change the section of (5) to get a copy of K/Z ( K ) × b Λ in h K, b Λ i . As noted above, this group is a subgroup of h K, Γ i , and so acts on Teich( S ).If K is positive-dimensional then so is K/Z ( K ), and so K/Z ( K ) contains a closedsubgroup isomorphic to a circle T . As the (possibly noneffective) action of b Λ on Teich( S )commutes with the action of T , we have that b Λ leaves Fix( T ) invariant. But this gives acontradiction, exactly as in Step 2 above, once we observe that [D] applies to b Λ, and so weobtain that K is trivial.To see that [D] applies to b Λ, there are two minor issues: her result is stated for Γ =Mod( S ), while we need the theorem for finite extensions and finite index subgroups of Γ.For the first issue we simply note that, just as mentioned in the first sentence of the proofof Theorem 26 in [BKK], the groups for which the theorem in [D] holds are closed underfinite extensions. The proof that [D] holds not just for Mod( S ), but for any finite indexsubgroup Γ ′ of Mod( S ), is verbatim the same as in [D], replacing the “Mess subgroups” B g constructed there with B g ∩ Γ ′ , which has finite index in B g . The two key properties of B g are: • B g is the fundamental group of a closed, triangulable topological manifold of dimension4 g −
5, and 7
The natural “point pushing” subgroup of B g is a closed surface group.Each of these properties is clearly preserved by taking finite index subgroups, and so theproof of the main theorem of [D] goes through when Mod( S ) replaced by any finite indexsubgroup Γ ′ . Step 4 ( I is trivial): By the previous steps, we know that Γ is a lattice in thesemisimple Lie group I , and that I has no compact factors. The proof of Proposition 3.1of [FW] now gives that there is a finite index subgroup Γ ′ of Γ so thatΓ ′ ≈ Γ × Γ ′ / Γ . (6)To give an idea of the proof of (6) from [FW], we begin by considering the exact sequence1 → Γ → Γ → Γ / Γ → . (7)As mentioned above, the extension, (7) is determined by a representation ρ : Γ / Γ → Out(Γ ), and by a cohomology class in H (Γ / Γ , Z (Γ ) ρ ).One shows that the image of ρ actually lies in Out( I ), which is finite since I is semisim-ple with finite center. After replacing Γ by a finite index subgroup Γ ′ , one then gets that theresulting representation ρ is trivial. One can also choose Γ ′ so that Γ ∩ Γ ′ is torsion-free.Since this group is a lattice in the semisimple Lie group with finite center I , it has finitecenter; since Γ ∩ Γ ′ is torsion-free, its center is trivial, so that the pertinent H vanishes.We now claim that for any finite index subgroup Γ ′ < Mod( S ), if Γ ′ = A × B then either A or B is finite. To see this, note that any such Γ ′ contains a pseudo-Anosov homeomorphism f (for example take a sufficiently high power of any pseudo-Anosov in Mod( S )). Thecentralizer in Γ ′ (indeed in Mod( S )) of any power of f has Z as a finite index subgroup (see[I1], Lemma 8.13). But in a product of two infinite groups, it is easy to see that any elementhas some power whose centralizer does not contain Z as a finite index subgroup.Thus either Γ is finite or Γ ′ / Γ is finite. The latter possibility implies that Γ ′ , henceMod( S ), has a finite index subgroup which is isomorphic to a lattice in the semisimple Liegroup I . If I is nontrivial, then it must contain a noncompact factor (by Step 3). Thiswould then contradict the theorem of Ivanov (see, e.g., § S ) is isomorphic to a lattice in a noncompact semisimple Lie group. Thusit must be that either I is trivial, or Γ is finite. If the latter possibility were to occur, then I would be compact since Γ is a lattice in I by Step 1. But this would contradict Step 3. ⋄ Let τ be a symmetry of (Teich( S ) , d ), i.e. an isometric involution with an isolated fixedpoint. Let L = h Mod( S ) , τ i be the group generated by Mod( S ) and by τ .By Theorem 1.2(b), which we have already proven, [ L : Mod( S )] < ∞ . Thus the actionof τ on L by conjugation induces a commensuration of Mod( S ), i.e. an isomorphism betweentwo finite index subgroups. Since Mod( S ) is residually finite, we can pass to further finite8ndex subgroups so that neither contains the hyperelliptic involution. By a theorem ofIvanov (see Theorem 8.5A of [I2]), since genus( S ) ≥ H of Mod( S ) with conjugation by some element φ of the extended mapping class group Mod ± ( S ), the index 2 supergroup of Mod( S ) whichincludes an orientation-reversing homotopy class of homeomorphism.We now claim that there exists an infinite order element ψ ∈ Mod( S ) that commuteswith τ . Note that since the conjugation action of τ on H agrees with the conjugation actionof φ , it is enough to produce an infinite order element ψ ∈ H so that ψ commutes with φ .Given this claim, we complete the proof of the theorem as follows. We are given that τ has an isolated fixed point x ∈ Teich( S ). By Smith theory, Fix( τ ) is Z / Z acyclic; inparticular Fix( τ ) is connected. Since ψ is an infinite order mapping class, we have that ψ ( x ) = x , by proper discontinuity of the action of Mod( S ) on Teich( S ). But τ ( ψ ( x )) = ψ ( τ ( x )) = ψ ( x )so that τ also fixes ψ ( x ) = x . As Fix( τ ) is connected and fixes at least two distinct points,it must have positive dimension. This contradicts the fact that x is an isolated fixed pointof τ . Thus such a τ cannot exist, and we are done.We now prove the claim. First note that since τ = Id, conjugation by φ is the identityon some finite index subgroup H of Mod( S ). Now there exists N > T α about any simple closed curve α , we have T Nα ∈ H . For any twist T α and anyelement f ∈ Mod( S ), we have the well-known formula f T Nα f − = T Nf ( α ) . Since φ ∈ Mod( S ), we can apply this formula with f = φ , giving that T Nα = T Nφ ( α ) for allsimple closed curves α . Since any positive power of a Dehn twist about a curve determinesthat curve, we have that φ ( α ) = α for each α . It follows that either φ = Id or genus( S ) = 2and φ is the hyperelliptic involution; our assumption that genus( S ) > φ = Id, and so commutes with any element ψ ∈ Mod( S ). We then pick ψ ∈ H to have infinite order. So we can assume φ = Id and φ = Id.Now any element φ ∈ Mod ± ( S ) of order 2 is represented by a homeomorphism φ of order2 (by a theorem of Fenchel). We now assume that g = genus( S ) >
2. First suppose thatFix( φ ) is discrete. Then S two-fold branched covers S/ h φ i . Since g = genus( S ) ≥
3, theRiemann-Hurwitz formula easily implies that either genus( S/ h φ i ) > S/ h φ i . Either way, the quotient S/ h φ i admits a self-homeomorphism ψ whose mapping class has infinite order. After perhaps replacing ψ by a finite power of ψ ,we know that ψ lifts to a self-homeomorphism ψ of S with the property that, in Mod( S ) wehave ψ φ = φψ . By replacing ψ with an appropriate power if necessary, we may assumethat ψ lies in the finite index subgroup H .If Fix( φ ) is not discrete then φ is orientation-reversing and Fix( φ ) is a union of c > S ′ := S/ h φ i has genus( S ′ ) >
0, then S ′ admitsan infinite order self-homeomorphism, which we can then lift as above to obtain ψ . Ifgenus( S ′ ) = 0 then S ′ is planar. Picking the outermost curve gives S ′ the structure ofa disk with ( c −
1) open disks removed from its interior. Thus the Euler characteristic9 ( S ′ ) = 1 − ( c −
1) = 2 − c . Since S is obtained from S ′ by gluing 2 copies of S ′ alongits ( χ = 0) boundary, we have 2 − g = χ ( S ) = 4 − c so that c = g + 1. Since we areassuming g >
2, it is clear that S ′ has an infinite order self-homeomorphism, and we aredone as above. ⋄ By Theorem 1.2, [Isom(Teich( S )) : Mod( S )] < ∞ . We pass to the index 2 subgroupIsom + (Teich( S )) of orientation-preserving isometries of Teich( S ). Note that that any ele-ment f ∈ Isom + (Teich( S )) must have Fix( f ) of codimension at least 2. Let f ∈ Isom + (Teich( S ))with f Mod( S ) be given. Ivanov’s theorem on commensurations of Mod( S ) mentionedabove implies that the conjugation action of f on some characteristic finite index subgroup H ′ of Mod( S ) agrees on some finite index subgroup H ≤ H ′ with conjugation by someelement φ ∈ Mod( S ). By composing with φ , we may assume the conjugation action of f on H is trivial, i.e. that f centralizes H . As [Isom + (Teich( S )) : H ] < ∞ , it must be that f n ∈ H for some n >
1. Now consider the exact sequence1 → H → h H, f i → h f i / h f n i → H is centerless (e.g. since it is finite index in Mod( S ) and so contains a pair of independentpseudo-Anosovs) and since the action of f on H is trivial, it follows that (8) splits, so that f n = Id. By passing to a power of f if necessary, we may assume that f has order someprime p ≥
2. Hence Fix( f ) is Z /p Z -acyclic by Smith theory. Now Fix( f ) has codimensionat least 2, and so has dimension at most 2. It is also a manifold. Since dim(Fix( f )) ≤
2, itfollows that Fix( f ) is contractible. But, just as noted in Case (B) of Step 3 in § H in these cases is not the fundamental group of a (closed or open) surface; it is also not thefundamental group of a 1-manifold by the same argument. We thus have a contradiction,so that f must be trivial. ⋄ References [BLM] J. Birman, A. Lubotzky, and J. McCarthy, Abelian and solvable subgroups of themapping class group,
Duke Math. Jour. , Vol.50, No.4 (1983), p.1107-1120.[BKK] M. Bestvina, M. Kapovich and B. Kleiner, Van Kampen’s embedding obstruction fordiscrete groups,
Inventiones Math.
Introduction to compact transformation groups , Pure and Applied Math.,Vol. 46. Academic Press, 1972.[Bro] K. Brown,
Cohomology of Groups , GTM Vol. 87, Springer-Verlag, 1982.[D] Z. Despotovic, Action dimension of mapping class groups, preprint.[DH] S. Deng and Z. Hou, The group of isometries of a Finsler space,
Pacific J. Math. DK] J.J. Duistermaat and J.A.C. Kolk,
Lie groups , Universitext. Springer-Verlag, Berlin,2000.[EK] C. Earle and I. Kra, On isometries between Teichm¨uller spaces,
Duke Math. J.
Annals of Math. ,Vol. 186, No.3, p.915-940 (2008).[H] J. Hausmann, Homology sphere bordism and Quillen plus construction, in
AlgebraicK-theory , Evanston1976, Springer Lect. Notes 551, 170–181.[I1] N. Ivanov, Subgroups of Teichm¨uller modular groups, translated from the Russian byE. J. F. Primrose and revised by the author, Translations of Math. Monographs, 115,AMS, 1992.[I2] N.V. Ivanov, Mapping class groups, in
Handbook of geometric topology , 523–633, 2002.[Mc] C. McMullen, The moduli space of Riemann surfaces is Kahler hyperbolic,
Annals ofMath. (2) 151 (2000), no. 1, 327–357.[Ro] H.L. Royden, Automorphisms and isometries of Teichmller space, in
Advances in theTheory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) pp. 369–383, Ann.of Math. Studies, No. 66., Princeton Univ. Press.pp. 369–383, Ann.of Math. Studies, No. 66., Princeton Univ. Press.