Hyperbolic Graphs of Surface Groups Honglin Min Rutgers University - Newark May 2, 2008
Notations Let S be a closed hyperbolic surface. Let T (S) be the Teichmuller space of S. Let PML be the projective measured lamination space of S. Let MCG(S) be the mapping class group of S. Let φ : S S be a pseudo-anosov homeomorphism, and let Φ MCG(S) be the mapping class of φ. Let Λ s, Λ u PML be the stable and unstable measured geodesic laminations of Φ. Let H be a subgroup of a group G. The subgroup V C(H) = {g G : finite index subgroup H < H, s.t., ghg 1 = h, h H } is called the virtual centralizer of H. Let Λ s, Λ u PML be the fixed points of a pseudo-anosov mapping class Φ, and let F ix{λ s, Λ u } denote the subgroup in MCG(S) whose elements fix Λ s, Λ u point wise. It is well-known that F ix{λ s, Λ u } = V C Φ. 1
Combination Theorem The fundamental group of the mapping torus of a pseudo-anosov homeomorphism of an oriented closed hyperbolic surface is hyperbolic. This was first proved by Thurston. A direct proof was given by Bestvina and Feighn. The Combination Theorem in their paper is an important tool to prove hyperbolicity of a graph of spaces. Theorem 1 (Bestvina and Feighn, Combination Theorem). Let SΓ be a finite graph of spaces with hyperbolic fundamental groups. Suppose that SΓ satisfies the quasi-isometrically embedded condition and the hallways flare condition, then the fundamental group of SΓ is hyperbolic. 2
A Hyperbolic-by-Hyperbolic hyperbolic group Theorem 2 (Mosher). Let S be an oriented closed hyperbolic surface, let Φ 1,, Φ m MCG(S) be an independent set of pseudo-anosov mapping classes of S, and let φ 1,, φ m Homeo(S) be the pseudo-anosov representatives of Φ 1,, Φ m respectively. If i 1,, i m are large enough positive integers, then the fundamental group of the graph of spaces K, as shown in Figure 1, is a hyperbolic group. S id φ i 2 2 φ i 1 1 S id S id φ im m S Figure 1: 3
By saying a set B of pseudo-anosov mapping classes is independent, we mean the sets F ix(φ) are pairwise disjoint for Φ B, where F ix(φ) consists of the attractor and the repeller of Φ on the space of projective measured laminations PML(S). Mosher applied the Combination Theorem as follows. Let B be a set of m independent pseudo-anosov mapping classes, then for sufficiently large integers i 1,, i m, the property 2m 1 out of 2m stretch holds for K. This implies that the hallways flare condition holds. Let φ i be the pseudo-anosov homeomorphism representatives of Φ i. We say the property 2m 1 out of 2m stretch holds if for any λ > 1, there exist sufficiently large integers i 1,, i m, such that for any closed geodesic path γ on S, γ must be stretched by a factor λ, by at least 2m 1 out of 2m elements {φ i 1 1, φ i 1 1, φ i m m, φ i m m }. 4
Hyperbolic graphs of surface groups A particular case studied here is: let G φ n be a graph of spaces as in Figure 2, where S and F are oriented φ n S S G φ n = p q F Figure 2: closed hyperbolic surfaces, p, q : S F are degree r covering maps, and φ is a pseudo-anosov homeomorphism of S. We shall find sufficient conditions under which the fundament group of G φ n is hyperbolic. 5
Let S 1 and S 2 be two copies of S equipped with the pullback metrics by the covering maps p and q respectively. There exist the derivative maps D p : P S 1 P F and D q : P S 2 P F of covering maps p and q respectively, where P S 1, P S 2 and P F are the projective tangent bundles of S 1, S 2 and F respectively. Let Λ s and Λ u denote the stable and unstable geodesic laminations of φ. Let T Λ s and T Λ u be the unit tangent vector spaces of Λ s and Λ u respectively. 6
Theorem 3. If D p T Λ s and D q T Λ u are injections, and their images are disjoint compact subsets of P F, then π 1 (G φ n) is a hyperbolic group, when n is large enough. More precisely, the hypothesis of the above theorem says the following. Let z be a point on F, let x i and y j be preimages of z under the map p, q respectively, that is, x i p 1 (z), y j q 1 (z), where i, j {1,, r}. If x i, x j Λ s, let L i, L j be the tangent lines of Λ s at x i, x j respectively. Similarly, if y i, y j Λ u, let V i, V j be the tangent lines of Λ u at y i, y j respectively. For each i, j {1,, r}, Dp(L i ) Dq(V j ) For each i j {1,, r}, Dp(L i ) Dp(L j ) For each i j {1,, r}, Dq(V i ) Dq(V j ) 7
Outline of the proof of Theorem 3 1. Let λ > 1. Given φ and 0 < ɛ < 1, there exist L > 0 and a positive integer N, for any geodesic segment α S, if α > L and α / N ɛ (Λ s ), then φ n (α) > λ α for n > N. N ɛ (Λ s ) denote all the hyperbolic geodesic segments β S which are in the ɛ neighborhood of the stabel geodesic lamination Λ s. Define a hyperbolic geodesic segment β is in the ɛ-neighborhood of a geodesic lamination Λ, if on a subset of β of length at least (1 ɛ)length(β), the distance between the tangent line of β and the set Λ, measured in P S, is at most ɛ. 8
2. We claim that under the hypothesis of Theorem 3, for any geodesic segment γ F, if γ is long enough, then at least 2r 1 out of 2r elements: { q(φ n p 1 1 (γ)),, q(φn p 1 r (γ)), p(φ n q1 1 (γ)),, p(φ n qr 1 (γ)) } are not less than λ γ, when n is large enough. 9
Applications Corollary 4. Let G, H be finite subgroups of M CG(S), and let Φ MCG(S) be a pseudo-anosov mapping class. If the virtual centralizer of Φ has trivial intersection with G and H, then G, Φ n HΦ n is a free product in MCG(S), i.e., G, Φ n HΦ n = G Φ n HΦ n, and its extension group is hyperbolic, for sufficiently large n. For any group homomorphism Q MCG(S), we obtain a group Γ and a commutative diagram of short exact sequences 1 π 1 (S, x) Γ Q 1 1 π 1 (S, x) MCG(S, x) MCG(S) 1 (1) Γ is called the extension group of Q. 10
Corollary 5. Suppose the virtual centralizer of Φ has trivial intersection with the deck transformation groups of p and q. In addition, suppose there are simple closed curves a F and c S, as shown in Figures 3 and 4, such that p 1 (a) = c, c q 1 (a), and q 1 (a) is disconnected. Then π 1 (G φ n) is hyperbolic if n is sufficiently large. There exists a pseudo-anosov homeomorphism φ 0 of S, such that π 1 (G φ n 0 ) is not abstractly commensurate to any surface-by-free group. α d c α β S Figure 3: a p(α) q(α) p(β) q(β) F Figure 4: 11
Outline of the proof of Corollary 5 Recall that, groups G and H are called abstractly commensurate, if there exist finite index subgroups G 1 < G and H 1 < H, so that G 1 is isomorphic to H 1. A group G is called a surface-by-free group, if there exists a hyperbolic surface or a hyperbolic orbifold S, and a free group K, such that there exists a short exact sequence: 1 π 1 (S) G K 1 Let G denote a graph of surfaces as in Figure 5, where S, S p q F Figure 5: F, p and q are as described in Corollary 5. We claim that π 1 (G) is not abstractly commensurate to a surface-by-free group according to the following lemma. 12
Lemma 6. Suppose the edge group π 1 (S) of π 1 (G) contains two nested sequences of finite index normal subgroups L 1 > L 2 > and R 1 > R 2 > which are constructed inductively as follows: 1. H 1 = p (π 1 (S)) q (π 1 (S)), L 1 = p 1 (H 1 ), R 1 = q 1 (H 1 ), G 1 = L 1 R 1 2. L i+1 = p 1 (H i+1 ), R i+1 = q 1 (H i+1 ) 3. G i+1 = L i+1 R i+1, H i+1 = p (G i ) q (G i ) IF L i R i for all i, then π 1 (G) is not abstractly commensurate to a surface-by-free group. We need to find a pseudo-anosov mapping class which fixes L i and R i for all i. First, we claim that if γ is a simple closed curve in S, such that [γ] L i for some i, then (τ γ ) of the Dehn-twist τ γ fixes L i. Second, notice that [α], [β] i (L i R i ). 13
Theorem 7 (Penner). Suppose that C and D are each disjointly embedded collections of essential simple closed curves (with no parallel components) in an oriented surface F so that C hits D efficiently and C D fills F. Let R(C +, D ) be the free semigroup generated by the Dehn twists {τ + c : c C} {τ 1 d : d D}. Each component map of the isotopy class of ω R(C +, D ) is either the identity or pseudo-anosov, and the isotopy class of ω is itself pseudo-anosov if each τ + c once in ω. and τ 1 d occur at least We need to show there exist disjointly embedded collections of essential simple closed curve C = α α, and D = β β, such that C D fills S. In addition, [α], [ α], [β] and [ β] i (L i R i ). 14
Theorem 8 (Bogopolski, Kudryavtseva,Zieschang). Let S be a closed orientable surface and g, h non-trivial elements of π 1 (S) both containing simple closed twosided curves γ and κ, resp. The group element h belongs to the normal closure of g if and only if h is conjugate to g ɛ or to (gug 1 u 1 ) ɛ, ɛ {1, 1}; here u is a homotopy class containing a simple closed curve µ which properly intersects γ exactly once. We know that the separating curve α as in Figure 3 represents an element in the normal closure of α, called N α. SinceN α is a subgroup of i (L i R i ), [α ], [β ] i (L i R i ). 15
Let ψ : S α S α be a pseudo-anosov homeomorphism. Let k be a large enough integer, such that ψ k (α ) β fills S α. Let α = ψ k (α ). Check there exists φ 0 constructed as the above, and the virtual centralizer of Φ 0 has trivial intersection with the deck transformation groups of p and q. 16
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