Trees and mapping class groups

TL;DR

This paper studies the properties of certain subgroups of the mapping class group related to punctured surfaces, proving their convex cocompactness and providing new proofs for existing theorems through geometric and group-theoretic methods.

Contribution

It establishes the convex cocompactness of finitely generated purely pseudo-Anosov subgroups in the kernel of a forgetful map, and offers new proofs of known theorems using novel approaches.

Findings

Finitely generated purely pseudo-Anosov subgroups are convex cocompact.

New proof of a theorem of I. Kra.

Relation between the kernel's action on the curve complex and actions on trees.

Abstract

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittlesey's group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of a theorem of J. Harer.