Geometry & Topology

Line patterns in free groups

Christopher H Cashen and Nataša Macura

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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. We also give an algorithm to detect such cut pairs.

Article information

Geom. Topol., Volume 15, Number 3 (2011), 1419-1475.

Received: 8 September 2010
Revised: 1 June 2011
Accepted: 29 June 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20E05: Free nonabelian groups

free group quasi-isometry rigidity line pattern Whitehead graph Whitehead's Algorithm


Cashen, Christopher H; Macura, Nataša. Line patterns in free groups. Geom. Topol. 15 (2011), no. 3, 1419--1475. doi:10.2140/gt.2011.15.1419.

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