Bounding genera of singular surfaces
Ultimate motivation: understand the curve complex for non-compact surfaces. But today, only closed surfaces around.
1. Genus distance
The curve complex of is hyperbolic, pseudo-Anosov diffeos act loxodromically, moving vertices at a linear speed.
Say if there is a compact surface with and a map mapping to and to . Let denote the equivalence class of . There are two cases. Either , i.e. is a separating curve. Or .
Fact. is 3-dense in (1-dense in the separating case).
Give, , define minimal genus of surface achieving . This is a metric. Morally, this is related to commutator length.
Question. Is this genus distance comparable to the distance in ?
2. Pseudo-Anosov distorsion
Example. Let be a pseudo-Anosov diffeo such that (such maps exist). Recall that . I claim that .
The proof requires 3-manifold topology. Let , achieved by some surface with a map . If is an essential simple closed curve, then is not null homotopic in (otherwise, one could surge into a surface of lower genus). Let be the mapping torus. It is hyperbolic. Let be the cyclic cover, diffeomorphic to , with periodic geometry, let be the deck transformation. Let be the closed geodesic in freely homotopic to . In ,
Thurston-Bonahon realise the composed map by a 1-Lipschitz map from equipped with a hyperbolic structure with concave boundary (it amounts to triangulating ). Note that Area is linear in . Also, the injectivity radius of is bounded from below independently on . So the diameter of is linear in . Thus
and . However, depends on and . Note that diameter is bounded.
Theorem 1 There is a constant , depending only on , such that for all ,
The proof is similar. Let minimize the genus of .
Fact. There exists a complete hyperbolic 3-manifold , homeomorphic to , with arbitrarily short representatives and .
It can be taken quasi-Fuchsian. Again, there exists a 1-Lipschitz map , where is hyperbolic with concave boundary, hence linear area. However, the injectivity radius of is not controlled. The thin part of is mapped to the thin part of , a union of solid tori. Let us electrify tubes: change to a metric which is zero on the thin part. Then remains 1-Lipschitz. In the electrified metric, the diameter of is bounded by its area, so the distance between and in the electrified metric is linear in .
This is a consequence of the Ending Lamination Theorem. One uses the quasi-isometric model of . The constant depends only on , but it is not effective.
Question (H. Wilton). Stable commutator length is more natural than commutator length. Would replacing and by powers simplify the argument ? I do not see how it could help.