Splitting line patterns in free groups

Cashen, Christopher H¹

¹Fakultät Für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Österreich

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 621-673

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.621

Classification : 20F65, 57M05, 20E05
Keywords: group splitting, line pattern, Whitehead graph, JSJ-decomposition, geometric word, virtually geometric multiword, relatively hyperbolic group, free group

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