Splitting Line Patterns in Free Groups

TL;DR

This paper constructs a boundary for finite rank free groups relative to certain conjugacy classes, and uses it to derive a JSJ decomposition, providing a new characterization of virtually geometric multiwords based on boundary planarity.

Contribution

It introduces a boundary construction for free groups relative to conjugacy classes and relates boundary properties to the group's JSJ decomposition and virtual geometricity.

Findings

The boundary construction captures the group's structure relative to conjugacy classes.

The quotient graph of groups corresponds to the JSJ decomposition.

Virtually geometric multiwords are characterized by boundary planarity.

Abstract

We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar.