Dynamics and geometry of surface homeomorphisms Harry Baik Berstein Seminar Nov 1, 2011 I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Geometric structures on general mapping tori I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Peano curves What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori What will be covered in this talk I Surfaces and mapping class groups I Automorphisms of tori I Geometric Structures on mapping tori I Geodesic laminations I Train tracks I Singular foliations I quadratic holomorphic differentials I Pseudo-Anosov automorphisms of surfaces I Geometric structures on general mapping tori I Peano curves Basic Definitions Surfaces A surface S is a 2-dimensional topological manifold. In this talk, they will be always assumed to be orientable, and often even closed. The theory I presented here is easily applied to the surfaces of finite type, ie., closed surfaces with finitely many points removed. The compact-open topology The compact-open topology on Map(X ; Y ) is the one generated by open sets of the form UK;U := fφ 2 Map(X ; Y )jφ(K) ⊂ Ug where K ⊂ X is compact and U ⊂ Y is open. The compact-open topology The group of self-homeomorphisms of S is denoted Homeo(S). We would like to think of Homeo(S) as a topological group. The topology came by regarding Homeo(S) as a subspace of Map(S; S) with compact-open topology. Smooth and PL category Without a proof, I state the following fact: The inclusions Diffeo(S) ,! Homeo(S); PL(S) ,! Homeo(S) induce isomorphisms on π0. So up to isotopy, we could work with whichever category is most convenient for our purposes. Essential loops and hierarchies Essential loops An embedded loop α ⊂ S is essential if it does not bound a disk or cobound an annulus together with a component of @S. A properly embedded arc β ⊂ S is essential if there is no other arc γ 2 @S such that β [ γ is an embedded circle which bounds a disk in S. Figure: This shows which are essential and which are not. Homotopy gives Isotopy Lemma Let α; β be essential loops or arcs in S. If α and β are properly homotopic, they are properly isotopic. Proof: We only show for loops. Whenever there is a bigon, remove it by isotopy. The we may assume that α and β intersect efficiently. Since α is essential, it is a nontrivial element in π1(S). Moreover any power of α is nontrivial too. Thus π1 has a subgroup ∼ < α >= Z. Consider the cover S^ of S corresponding to this subgroup. Note that S^ is a cylinder. Homotopy gives Isotopy Choose liftsα; ^ β^ of the loops α; β. Note that they intersect at the even number of points, since α ' β implies that their algebraic intersection number is 0. Figure: If they are disjoint (above), they are isotopic. Otherwise (below), we get a contradiction, since we get an embedded bigon. Hierarchy Let S be a connected surface of finite type and non-positive Euler characteristic. Let α1 be an essential loop or arc on S. Then we can get a new surface S1, possibly disconnected, by cutting S along α1. We achieve either smaller genus or bigger Euler characteristic. Figure: Cutting along the red separating curve gives us two once punctures torus T . Note χ(T ) = −1 > χ(Σ2) = −2. Cutting along the blue non-separating one gives a smaller genus surface. Hierarchy After finitely many such cuts, one is left a collection of disks and get a hierarchy α1 α2 αn S S1 ··· Sn = [i Di The Mapping Class Group MCG(S) The quotient group MCG(S) := Homeo(S)=Homeo0(S) is called the Mapping Class Group of S. MCG +(S) will denote the subgroup consisting of orientation-preserving homeomorphisms. Lemma Suppose φ1; φ2 2 Homeo(S) represents the same element in MCG(S). Then the induced (outer) automorphisms (φ1)∗; (φ2)∗ of π1(S) are equal. Proof Homotopic maps induce isomorphic actions on π1. It follows that there is an induced homomorphism ρ : MCG(S) ! Out(π1(S)). The Mapping Class Group Lemma Let φ : S ! S be a proper homotopy equivalence. If α is an essential loop or arc in S, then φ(α) is properly homotopic to an essential loop or arc. Proof Omitted. The Mapping Class Group Dehn-Nielsen Theorem Suppose S is a closed orientable surface of genus g ≥ 1. Then ρ is an isomorphism. Otherwise, ρ is an injection, with image equal to the subgroup of Out(π1(S)) consisting of automorphisms which permute the peripheral subgroups. Proof Omitted. Coarse Geometry Proper/geodesic metric spaces A complete metric space is proper if it is locally compact. A metric space is geodesic if any two points may be joined by an isometrically embedded arc. δ-hyperbolicity A geodesic metric space X is δ-hyperbolic for δ ≥ 0 if for all geodesic triangles pqr, every point on the edge pq is within distance δ from the union of the edges qr and rp. A geodesic space is Gromov hyperbolic if it is δ-hyperbolic for some δ. e.g. Trees are 0-hyperbolic. Coarse Geometry Morse Lemma Let X be a δ-hyperbolic space. Then for every k; there is a universal constant C(δ; k; ) such that every (k; )-quasigeodesic segment with endpoints p; q 2 X lies in the C-neighborhood of any geodesic joining p and q. Gromov boundary To a δ-hyperbolic proper metric space X , we may associate the ideal boundary @X , defined as follows. As a set, @X is a set of equivalence classes of quasigeodesic rays, where r ∼ r 0 if each is contained in the T-neighborhood of the other for some T (which might depend on r; r 0). Let [r] be an arbitrary equivalence class. Let x0 be any point in X . Then [r] contains a geodesic ray s which can be taken to satisfy s(0) = x0. Automorphisms of tori Let T denote the standard 2-dimensional torus. Then π1(S) = Z ⊕ Z. Since this groups is Abelian, we have an equality Out(π1(T )) = Aut(π1(T )) = GL(2; Z) provided that we pick a basis, and MCG+(T ) = SL(2; Z). (Since we have integral entries, the elements of GL(2; Z) has det = ±1, so the orientation preserving elements form SL(2; Z).) Let φ 2 MCG +(T ). Under this identification, φ corresponds to a 2 × 2 a b matrix where a; b; c; d 2 and det(φ) = 1. The c d Z −1 eigenvalues of φ are λ, λ for some λ 2 C, where tr(φ) := a + d = λ + λ−1. Automorphisms of tori Since the trace of φ is real, if λ, λ−1 are not real, they are both on the unit circle, and therefore jtr(φ)j < 2. Since a and d are integers, in this case we have tr(φ) = 0; ±1. a b When tr(φ) = 0, φ = with a2 + bc = −1.