Splitting line patterns in free groups

Cashen, Christopher H

doi:10.2140/agt.2016.16.621

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Splitting line patterns in free groups

Christopher H Cashen

Algebraic & Geometric Topology 16 (2016) 621–673

DOI: 10.2140/agt.2016.16.621

Abstract

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.

Keywords

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

References

Publication

Received: 30 September 2010

Revised: 26 December 2015

Accepted: 12 January 2016

Published: 26 April 2016

Authors

Christopher H Cashen
Fakultät Für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 1090 Wien Österreich
http://www.mat.univie.ac.at/~cashen

Volume 16, issue 2 (2016)