The Michigan Mathematical Journal

Nielsen Realization by Gluing: Limit Groups and Free Products

Sebastian Hensel and Dawid Kielak

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We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT(0).

The proofs rely on a new version of Stallings’ theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained.

Article information

Michigan Math. J., Volume 67, Issue 1 (2018), 199-223.

Received: 19 September 2016
Revised: 22 August 2017
First available in Project Euclid: 20 February 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations


Hensel, Sebastian; Kielak, Dawid. Nielsen Realization by Gluing: Limit Groups and Free Products. Michigan Math. J. 67 (2018), no. 1, 199--223. doi:10.1307/mmj/1519095620.

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