Line Patterns in Free Groups

TL;DR

This paper investigates the topology of line patterns in free groups through decomposition spaces, revealing conditions under which quasi-isometries act as isometries based on the presence of cut pairs.

Contribution

It establishes a connection between the topological properties of decomposition spaces and the isometric actions of quasi-isometries in free groups.

Findings

Quasi-isometries preserve line patterns if no cut pairs exist in the decomposition space.

The topology of the decomposition space determines the isometric nature of quasi-isometry actions.

Conditions for isometry actions are characterized by the absence of cut pairs in the boundary topology.

Abstract

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space.