Notes of Brian Bowditch’s Cambridge lecture 20-04-2017

Bounding genera of singular surfaces

Ultimate motivation: understand the curve complex for non-compact surfaces. But today, only closed surfaces {\Sigma} around.

1. Genus distance

The curve complex {\mathcal{C}} of {\Sigma} is hyperbolic, pseudo-Anosov diffeos act loxodromically, moving vertices at a linear speed.

Say {\alpha\sim\beta} if there is a compact surface {S} with {\partial S=\alpha_0\cup \beta_0} and a map {S\rightarrow\Sigma} mapping {\alpha_0} to {\alpha} and {\beta_0} to {\beta}. Let {A=A(\alpha)} denote the equivalence class of {\alpha}. There are two cases. Either {\alpha\sim0}, i.e. {\alpha} is a separating curve. Or {\alpha\not\sim 0}.

Fact. {A} is 3-dense in {\mathcal{C}} (1-dense in the separating case).

Give, {\alpha,\beta\in A}, define {\rho(\alpha,\beta)=} minimal genus of surface {S} achieving {\alpha\sim\beta}. This is a metric. Morally, this is related to commutator length.

Question. Is this genus distance comparable to the distance in {\mathcal{C}}?

2. Pseudo-Anosov distorsion

Example. Let {\phi} be a pseudo-Anosov diffeo such that {h(\alpha)\sim\alpha} (such maps exist). Recall that {d(\alpha,\phi^n\alpha)\sim n}. I claim that {\rho(\alpha,\phi^n\alpha)\sim n}.

The proof requires 3-manifold topology. Let {g=\rho(\alpha,\phi^n\alpha)}, achieved by some surface {S} with a map {f:S\rightarrow\Sigma}. If {\gamma\subset S} is an essential simple closed curve, then {f(\gamma} is not null homotopic in {\Sigma} (otherwise, one could surge {S} into a surface of lower genus). Let {M_\phi} be the mapping torus. It is hyperbolic. Let {M\rightarrow M_\phi} be the cyclic cover, diffeomorphic to {\Sigma\times{\mathbb R}}, with periodic geometry, let {\psi} be the deck transformation. Let {\alpha^*} be the closed geodesic in {M} freely homotopic to {\alpha}. In {M},

\displaystyle  \begin{array}{rcl}  d(\alpha^*,\psi^n\alpha^*)\sim n. \end{array}

Thurston-Bonahon realise the composed map {F:S\rightarrow\Sigma\rightarrow M} by a 1-Lipschitz map from {S} equipped with a hyperbolic structure with concave boundary (it amounts to triangulating {S}). Note that Area{(S)\leq 2\pi(2g+1)} is linear in {g}. Also, the injectivity radius of {S} is bounded from below independently on {n}. So the diameter of {S} is linear in {g}. Thus

\displaystyle  \begin{array}{rcl}  d(\alpha^*,\psi^n\alpha^*)\leq C\,g, \end{array}

and {g\geq c\, n}. However, {c} depends on {\phi} and {\alpha}. Note that diameter{(\alpha^*)} is bounded.

3. Result

Theorem 1 There is a constant {L}, depending only on {\Sigma}, such that for all {\alpha,\beta\in A},

\displaystyle  \begin{array}{rcl}  d(\alpha,\beta)\leq L\,\rho(\alpha,\beta). \end{array}

The proof is similar. Let {f:S\rightarrow\Sigma} minimize the genus {g} of {S}.

Fact. There exists a complete hyperbolic 3-manifold {M}, homeomorphic to {\Sigma\times{\mathbb R}}, with arbitrarily short representatives {\alpha^*} and {\beta^*}.

It can be taken quasi-Fuchsian. Again, there exists a 1-Lipschitz map {F:S\rightarrow\Sigma\rightarrow M}, where {S} is hyperbolic with concave boundary, hence linear area. However, the injectivity radius of {M} is not controlled. The thin part of {S} is mapped to the thin part of {M}, a union of solid tori. Let us electrify tubes: change to a metric which is zero on the thin part. Then {F} remains 1-Lipschitz. In the electrified metric, the diameter of {S} is bounded by its area, so the distance between {\alpha^*} and {\beta^*} in the electrified metric is linear in {g}.

Fact. {d(\alpha,\beta)\leq L\,d_{elec}(\alpha^*,\beta^*)}.

This is a consequence of the Ending Lamination Theorem. One uses the quasi-isometric model of {M}. The constant {L} depends only on {\Sigma}, but it is not effective.

Question (H. Wilton). Stable commutator length is more natural than commutator length. Would replacing {\alpha} and {\beta} by powers simplify the argument ? I do not see how it could help.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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