Kirby Problem 2.18

Definitions and mathematical statement

Let $S=S_{g,n}$ be an oriented finite-type surface of genus $g$ with $n$ punctures and negative Euler characteristic, supporting a pseudo-Anosov, and let $\phi:S\to S$ be pseudo-Anosov: it preserves a pair of transverse measured singular foliations, multiplying their transverse measures by $\lambda$ and $\lambda^{-1}$ for some $\lambda>1$. Its mapping torus $M_\phi=(S\times[0,1])/((x,1)\sim(\phi(x),0))$ has a complete finite-volume hyperbolic metric of curvature $-1$; write $V(\phi)$ for its volume.

Two relevant metric spaces are the Teichmüller space $\mathcal T(S)$ with its Weil--Petersson metric (the $L^2$ metric on harmonic Beltrami differentials), and the pants graph $\mathcal P(S)$. Vertices of the latter are maximal disjoint essential multicurves. An edge replaces one curve by one meeting it minimally in the complementary one-holed torus or four-holed sphere. For either space $Y$, define stable translation length by \[ \tau_Y(\phi)=\lim_{n\to\infty}\frac{d_Y(y,\phi^n y)}n, \] which is independent of $y$.

Find strong explicit lower bounds for $V(\phi)$ in terms of these translation lengths. In particular, determine effective constants, and their dependence on $(g,n)$, in inequalities $V(\phi)\geq c_{g,n}\tau_Y(\phi)$ valid for every pseudo-Anosov $\phi$. The problem seeks quantitative bounds, beyond a comparison with unspecified constants for each fixed surface.

Short English statement

Give effective lower bounds on the volume of a hyperbolic surface bundle from how far its monodromy moves points in Weil--Petersson space or the pants graph.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2766 (KP-2.18), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.18. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.