[Paper Review] Invariant geodesics in the curve complex under point-pushing pseudo-Anosov mapping classes

Chaohui Zhang|arXiv (Cornell University)|Mar 16, 2013

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper identifies point-pushing pseudo-Anosov mapping classes on a punctured Riemann surface of genus >1 that preserve at least one bi-infinite geodesic in the curve complex. Using geometric and topological techniques involving curve configurations and isometric actions, the author proves that such invariant geodesics exist precisely when the associated filling curve intersects the puncture-encircling curve exactly once, establishing a complete characterization of these invariant geodesics in terms of intersection number properties.

ABSTRACT

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

Motivation & Objective

• To determine which point-pushing pseudo-Anosov mapping classes on a punctured surface preserve at least one bi-infinite geodesic in the curve complex.
• To characterize the geometric and topological conditions under which such invariant geodesics exist.
• To establish a bijection between certain curves on the filled surface and invariant bi-infinite geodesics in the curve complex of the punctured surface.
• To prove that the existence of invariant geodesics is equivalent to the associated filling curve intersecting the puncture-encircling curve exactly once.

Proposed method

• The paper analyzes the action of point-pushing pseudo-Anosov maps on the curve complex using configurations of curves and their lifts to the universal cover.
• It employs the concept of axis-aligned configurations (τ, Ω, U) to track the dynamics of curve images under iterated mapping class action.
• The proof relies on the uniqueness of geodesic segments in the curve complex, established via isometric invariance and contradiction arguments.
• It uses the fact that the mapping class group acts isometrically on the curve complex to show that the concatenation of forward and backward geodesic segments forms a bi-infinite geodesic.
• The argument distinguishes cases based on the intersection number between the filling curve and the puncture-encircling curve, focusing on the case where it is exactly one.
• It proves injectivity of the correspondence between curves and geodesics by showing that distinct curves on the filled surface yield disjoint geodesics in the curve complex.

Experimental results

Research questions

• RQ1 Which point-pushing pseudo-Anosov mapping classes on a punctured surface preserve a bi-infinite geodesic in the curve complex?
• RQ2 What topological or geometric condition on the associated filling curve guarantees the existence of such invariant geodesics?
• RQ3 Is the correspondence between curves on the filled surface and invariant geodesics in the curve complex of the punctured surface injective and well-defined?
• RQ4 Under what conditions does the orbit of a curve under a point-pushing pseudo-Anosov map form a bi-infinite geodesic in the curve complex?
• RQ5 Can the existence of invariant geodesics be characterized purely by the intersection number between the filling curve and the puncture-encircling curve?

Key findings

• A point-pushing pseudo-Anosov map preserves a bi-infinite geodesic in the curve complex if and only if the associated primitive, oriented, filling curve on the filled surface intersects the puncture-encircling curve exactly once.
• For such maps, the orbit of a curve under iteration forms a unique bi-infinite geodesic, and this geodesic is preserved by all powers of the mapping class.
• The correspondence between curves on the filled surface (with intersection number one with the puncture curve) and invariant geodesics in the curve complex is well-defined and injective.
• The proof shows that any two such geodesics arising from distinct curves on the filled surface are disjoint in the curve complex.
• The uniqueness of geodesic segments connecting iterated images ensures that the full orbit forms a geodesic path, confirming the bi-infinite geodesic structure.
• The result provides a complete characterization of invariant geodesics for this class of pseudo-Anosov maps, contrasting with the general case where only powers of such maps preserve finitely many geodesics.