Line Patterns in Free Groups
Christopher H. Cashen, Natasa Macura

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.
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