Line Patterns in Free Groups

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.