Line Patterns in Free Groups

Abstract

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space.

35 pages, 22 figures, PDFLatex; v2. finite index requires extra hypothesis; v3. 37 pages, 24 figures: updated references and add example in Section 6.3 of a rigid pattern for which the free group is not finite index in the group of pattern preserving quasi-isometries; v4. 40 pages, 26 figures: improved exposition and add example in Section 6.4 of a rigid pattern whose cube complex is not a tree

References in corpus (2)

• Quasi-isometric classification of graph manifold groups
• Quasi-isometries Between Tubular Groups

Cited by in corpus (8)

• Quasi-isometries Between Groups with Two-Ended Splittings
• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• One-ended subgroups of graphs of free groups with cyclic edge groups
• Splitting Line Patterns in Free Groups
• Virtual Geometricity is Rare
• Computing JSJ decompositions of hyperbolic groups
• Immersed cycles and the JSJ decomposition
• RAAGedy right-angled Coxeter groups II: in the quasiisometry class of the tree RAAGs