## The Michigan Mathematical Journal

### Nielsen Realization by Gluing: Limit Groups and Free Products

#### Abstract

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

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

Dates
Revised: 22 August 2017
First available in Project Euclid: 20 February 2018

https://projecteuclid.org/euclid.mmj/1519095620

Digital Object Identifier
doi:10.1307/mmj/1519095620

Mathematical Reviews number (MathSciNet)
MR3770860

Zentralblatt MATH identifier
06965596

#### Citation

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. https://projecteuclid.org/euclid.mmj/1519095620

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