Combinations of parabolically geometrically finite groups and their geometry

TL;DR

This paper investigates parabolically geometrically finite (PGF) subgroups within mapping class groups, establishing a combination theorem, constructing new PGF examples, and demonstrating their undistorted nature in the larger group.

Contribution

It introduces a combination theorem for PGF groups using subsurface projection, expanding the understanding of their geometry and providing practical methods for application.

Findings

Proved a combination theorem for graphs of PGF groups.

Constructed new examples of PGF groups.

Showed PGF groups are undistorted in mapping class groups.

Abstract

In this paper, we study the class of parabolically geometrically finite (PGF) subgroups of mapping class groups, introduced by Dowdall-Durham-Leininger-Sisto. We prove a combination theorem for graphs of PGF groups (and other generalizations) by utilizing subsurface projection to obtain control on the geometry of fundamental groups of graphs of PGF groups, generalizing and strengthening methods of Leininger-Reid. From this result, we construct new examples of PGF groups and provide methods for how to apply the combination theorem in practice. We also show that PGF groups are undistorted in their corresponding mapping class group.