Cashen, Christopher H. Splitting line patterns in free groups. (English) Zbl 1383.20024 Algebr. Geom. Topol. 16, No. 2, 621-673 (2016). Summary: 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. Cited in 1 ReviewCited in 7 Documents MSC: 20F65 Geometric group theory 57M05 Fundamental group, presentations, free differential calculus 20E05 Free nonabelian groups 20E45 Conjugacy classes for groups 20E08 Groups acting on trees Keywords:group splitting; line pattern; Whitehead graph; JSJ-decomposition; geometric word; virtually geometric multiword; relatively hyperbolic group; free group PDFBibTeX XMLCite \textit{C. H. Cashen}, Algebr. Geom. Topol. 16, No. 2, 621--673 (2016; Zbl 1383.20024) Full Text: DOI arXiv Link