[Paper Review] On the minimum of asymptotic translation lengths of point-pushing pseudo-Anosov maps on one punctured Riemann surfaces

Chaohui Zhang|arXiv (Cornell University)|Jan 21, 2013

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper proves that the minimum asymptotic translation length of point-pushing pseudo-Anosov maps on a one-punctured Riemann surface of genus $ p > 1 $ is exactly 1. Using curve complex geometry and hyperbolic dynamics, the author establishes a sharp lower bound via configuration analysis of geodesics and their images under iterated mapping classes, showing that the translation length cannot be less than 1 and is achieved in specific geometric settings.

ABSTRACT

We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.

Motivation & Objective

• To determine the infimum of asymptotic translation lengths for point-pushing pseudo-Anosov maps on one-punctured Riemann surfaces of genus $ p > 1 $.
• To establish a sharp lower bound for $ L_{ ilde{\mathcal{C}}}({\mathscr{F}}) $, the minimum translation length over the point-pushing subgroup $ \mathscr{F} $.
• To contrast the behavior of $ L_{\mathcal{C}}({\mathscr{F}}) $ with that of $ L_{\mathcal{C}}(\mathrm{Mod}(S)) $ and $ L_{\mathcal{C}}(\mathscr{N}_k(S)) $, which decay as genus increases.
• To prove that the asymptotic translation length is exactly 1, independent of genus, for the point-pushing subgroup.
• To provide a geometric characterization of when equality $ d_{\mathcal{C}}(u, f^m(u)) = |m| $ is achieved, linking it to intersection properties and level configurations in the universal cover.

Proposed method

• Utilizes the curve complex $ \mathcal{C}(S) $ equipped with a path metric $ d_{\mathcal{C}} $, where vertices correspond to simple closed geodesics.
• Analyzes the action of point-pushing pseudo-Anosov maps $ f \in \mathscr{F} $, which are induced by pushing a puncture along a filling geodesic $ \tilde{c} $ on the closed surface $ \tilde{S} $.
• Applies hyperbolic geometry by lifting the surface to the universal cover $ \mathbf{H} $, where the mapping class corresponds to a hyperbolic isometry $ g $ with an axis.
• Employs a configuration-based argument using horoballs $ \Omega_j $, maximal sets $ \Delta_j^* $, and level structures in the universal cover to control distances in the curve complex.
• Uses the triangle inequality and inductive control of distances $ d_{\mathcal{C}}(u, f^m(u)) $ to show $ \tau_{\mathcal{C}}(f) \leq 1 $, and proves $ \tau_{\mathcal{C}}(f) \geq 1 $ via contradiction and level analysis.
• Applies results from [12], [15], and [16] on horoball configurations and intersection numbers to establish that $ d_{\mathcal{C}}(u, f^m(u)) \geq |m| $ for $ |m| \geq 3 $, implying $ \tau_{\mathcal{C}}(f) \geq 1 $.

Experimental results

Research questions

• RQ1 What is the infimum of asymptotic translation lengths for point-pushing pseudo-Anosov maps on a one-punctured surface of genus $ p > 1 $?
• RQ2 How does the asymptotic translation length of the point-pushing subgroup $ \mathscr{F} $ compare to that of the full mapping class group or lower central series subgroups?
• RQ3 Under what geometric conditions does $ d_{\mathcal{C}}(u, f^m(u)) = |m| $ hold for a pseudo-Anosov map $ f \in \mathscr{F} $?
• RQ4 Can the asymptotic translation length be bounded below by 1, and is this bound sharp for the point-pushing subgroup?
• RQ5 What role do the level structures of horoballs and their maximal covers play in controlling distances in the curve complex?

Key findings

• The minimum asymptotic translation length $ L_{\mathcal{C}}(\mathscr{F}) $ for the point-pushing subgroup $ \mathscr{F} $ on a one-punctured surface of genus $ p > 1 $ is exactly 1.
• For any pseudo-Anosov map $ f \in \mathscr{F} $, there exists a vertex $ u \in \mathcal{C}_0(S) $ such that $ d_{\mathcal{C}}(u, f^m(u)) \geq |m| $ for all integers $ |m| \geq 3 $, implying $ \tau_{\mathcal{C}}(f) \geq 1 $.
• The upper bound $ \tau_{\mathcal{C}}(f) \leq 1 $ is established via the triangle inequality and the fact that $ d_{\mathcal{C}}(u, f^{2m}(u)) \leq 2m $, with equality in distance growth along the orbit.
• The equality $ d_{\mathcal{C}}(u, f^m(u)) = |m| $ holds if and only if the initial geodesic $ u $ intersects the filling curve $ \tilde{c} $ exactly once and all intermediate horoball configurations lie at their respective levels.
• The bound of 1 is optimal and cannot be improved, as shown by the existence of configurations achieving equality in the distance growth.
• The result contrasts sharply with the full mapping class group, where $ L_{\mathcal{C}}(\mathrm{Mod}(S)) \to 0 $ as genus increases, while $ L_{\mathcal{C}}(\mathscr{F}) = 1 $ remains constant.