Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

TL;DR

This paper classifies certain infinite graphs based on their pure mapping class groups, analyzing their algebraic structures and properties to advance understanding of their geometric and algebraic rigidity.

Contribution

It provides a complete classification of locally finite infinite graphs with pure mapping class groups that have coarsely bounded generating sets, and studies their algebraic properties.

Findings

Classification of graphs with coarsely bounded generating sets

Semidirect product decomposition of groups

Conditions for residual finiteness and Tits alternative

Abstract

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.