DRILLING SURFACES AND SURFACE-AUTOMORPHISMS KAZUHIRO ICHIHARA AND KIMIHIKO MOTEGI

DRILLING SURFACES AND SURFACE-AUTOMORPHISMS

KAZUHIRO ICHIHARA AND KIMIHIKO MOTEGI

In the following, let F be an orientable closed surface and f : F → F an orientation pre- serving diffeomorphism which leaves some point x and its diskal neighborhood D x invariant as a set. Let us denote F − intD ^x by ¯ F and f | F ^¯ by ¯ f . Similarly ¯ φ will mean a restriction φ| F ^¯ for such a diffeomorphism φ.

It is well-known that any self-diffeomorphism of a surface is isotopic to be either periodic, reduced, or pseudo-Anosov. See [5] for details. Among them, it is weidely believed that pseudo-Anosov ones are most common. For example, see [6].

Suppose that f is not isotopic to a pseudo-Anosov diffeomorphism. Then after drilling the surface, can we obtain the restriction ¯ f : ¯ F → F ¯ which is isotopic to a pseudo-Anosov diffeomorphism?

The most simplest case may be the case that f is isotopic to the identity map, which was studied by Kra [4] from a viewpoint of Teichm¨ uller space theory. To make precise we consider the following setting. Suppose that f is isotopic to the identity. Then there is an isotopy J of f to the identity: J : I × F → F, J(0, x) = f (x) = x and J (1, x) = x. By putting c(t) = J (t, x), we obtain an oriented closed curve c on F with a base point x. Let us denote such a map f by σ c .

Theorem 1 (Kra). A diffeomorphism σ ¯ ^c : ¯ F → F ¯ is isotopic to a pseudo-Anosov diffeo- morphism for any stably filling curve c (i.e., any curve freely homotopic to c intersects every essential embedded loop on F ).

This result was originally proven by Kra [4]. We will give an alternative proof of Theorem 1 based on 3-manifold topology. Our proof is quite different from that of [4].

More generally, for a given diffeomorphism f with f (x) = x and f (D ^x ) = D x , we ask:

can we isotope f to a diffeomorphism g : F → F with g(x) = x, g(D ^x ) = D ^x so that

¯

g is isotopic to a pseudo-Anosov diffeomorphism? (During the isotopy x and D ^x are not necessarily invariant.)

A result of Birman [1, 2] enables us to ask more precisely as follows.

Question. For which curve c (with a base point x), can σ ¯ c ◦ f ¯ be isotopic to a pseudo- Anosov diffeomorphism?

Note that Theorem 1 answers to this question in the case that f is the identity map.

We will consider the question for irreducible maps.

Theorem 2. Let f : F → F be an irreducible diffeomorphism satisfying f (x) = x and f (D ^x ) = D ^x for some x ∈ F .

Resume for symposium ‘

II’ (Hyperbolic Spaces and Discrete Groups II), 2001.12.5(Wed), 16

30-17

30.

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2 KAZUHIRO ICHIHARA AND KIMIHIKO MOTEGI

(1) If f is not isotopic to a periodic diffeomorphism (and hence f is isotopic to a pseudo-Anosov diffeomorphism), then σ ¯ c ◦ f ¯ is a pseud-Anosov diffeomorphism for any curve c on F .

(2) If f is isotopic to a periodic diffeomorphism g without fixed point, then σ ¯ ^c ◦ f ¯ is a pseudo-Anosov diffeomorphism for any curve c.

(3) If f is a periodic diffeomorphism with a single fixed point x, then σ ¯ c ◦ f ¯ is a pseudo- Anosov diffeomorphism for any curve c such that [c] 6= α ^{− 1} f _∗ (α) for any element α ∈ π 1 (F, x).

Remark that, in Theorem 2 (3), ¯ σ c ◦ f ¯ is periodic if [c] = α ^{− 1} f _∗ (α) for some α ∈ π 1 (F, x).

Also we show that there are plenty of stably filling curves and curves satisfying the condition in Theorem 2 (3).

We then provide an application motivated by a question about Dehn surgery on knots.

Let K be a knot in the 3-sphere or a homology 3-sphere. We denote the exterior of K by E(K). A slope γ on ∂E(K) (i.e., ∂E(K)-isotopy class of essential, simple closed curves) is called a boundary slope if there is an essential surface F in E(K) such that ∂F ∩ ∂E(K) is a nonempty set of parallel simple closed curves of slope γ. A Seifert fiber space is said to be small if it admits a Seifert fibration over S ² with at most three exceptional fibers.

Among boundary slopes, only a longitudinal slope can be a slope along which a surgery yields a small Seifert fiber space. On the other hand, there is no known hyperbolic knots in S ³ on which longitudinal surgeries yield Seifert fiber spaces. In [3, 5.16] Teragaito conjectures that a longitudinal surgery on any hyperbolic knot in S ³ can never produce a Seifert fiber space.

We apply Theorem 2 to obtain the following weaker result.

Theorem 3. For any integer g ≥ 2, there is a homology 3-sphere M and a hyperbolic, fibered knot K of genus g in M on which longitudinal surgery yields a small Seifert fiber space.

References

1. J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math.

22 (1969), 213–238.

2. J. S. Birman; Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J., 1974.

3. Problems in Low dimensional topology (in Japanese), edited by H.Goda 1996.

4. I. Kra; On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta mathematica 146 (1981), 231–270.

5. J.-P. Otal, The hyperbolization theorem for fibered 3-manifolds, Translated from the 1996 French original by Leslie D. Kay. SMF/AMS Texts and Monographs, 7. American Mathematical Society, Providence, RI; Societe Mathematique de France, Paris, 2001.

6. M. Takasawa, Enumeration of mapping classes for the torus, Geom. Dedicata 85 (2001), no. 1-3, 11–19.

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O- okayama 2–12–1, Meguro-ku, Tokyo 152–8552, Japan.

E-mail address : [email protected]

Department of Mathematics, Nihon University, 3–25–40 Sakurajosui, Setagaya-ku, Tokyo 156–

8550, Japan.

E-mail address : [email protected]