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Splitting line patterns in free groups

Christopher H Cashen

Algebraic & Geometric Topology 16 (2016) 621–673

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.

group splitting, line pattern, Whitehead graph, JSJ-decomposition, geometric word, virtually geometric multiword, relatively hyperbolic group, free group
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 57M05, 20E05
Received: 30 September 2010
Revised: 26 December 2015
Accepted: 12 January 2016
Published: 26 April 2016
Christopher H Cashen
Fakultät Für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
1090 Wien