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Splitting line patterns in free groups. (English) Zbl 1383.20024

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.

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
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