Splitting Line Patterns in Free Groups

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.Comment: 22 pages, 6 figures; v2 fixed a few typos; v3 38 pages, 21 figures; v4 30 pages, 11 figures 'Preliminaries' section expanded to make paper self-contained and split into two sections. Some arguments refactored and simplified. Paper streamlined; v5 56 pages, 21 figures Added examples and improved exposition according to referee comments. To appear in Algebraic & Geometric Topolog