The Michigan Mathematical Journal

Nielsen Realization by Gluing: Limit Groups and Free Products

Sebastian Hensel and Dawid Kielak

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1519095620

Digital Object Identifier
doi:10.1307/mmj/1519095620

Mathematical Reviews number (MathSciNet)
MR3770860

Zentralblatt MATH identifier
06965596

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

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


Export citation

References

  • [AB] E. Alibegović and M. Bestvina, Limit groups are $\mathrm{CAT}(0)$, J. Lond. Math. Soc. (2) 74 (2006), 259–272.
  • [BH] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999.
  • [Bro] S. Brown, A gluing theorem for negatively curved complexes, J. Lond. Math. Soc. (2) 93 (2016), 741–762.
  • [BKM] I. Bumagin, O. Kharlampovich, and A. Miasnikov, The isomorphism problem for finitely generated fully residually free groups, J. Pure Appl. Algebra 208 (2007), 961–977.
  • [Cul] M. Culler, Finite groups of outer automorphisms of a free group, Contributions to group theory, Contemp. Math., 33, pp. 197–207, Amer. Math. Soc., Providence, RI, 1984.
  • [CV] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119.
  • [Dun1] M. J. Dunwoody, Cutting up graphs, Combinatorica 2 (1982), 15–23.
  • [Dun2] M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), 449–457.
  • [Gru] I. Gruschko, Über die Basen eines freien Produktes von Gruppen, Rec. Math. [Mat. Sbornik] N.S. 8(50) (1940), 169–182.
  • [GL] V. Guirardel and G. Levitt, The outer space of a free product, Proc. Lond. Math. Soc. (3) 94 (2007), 695–714.
  • [HM] M. Handel and L. Mosher, Relative free splitting and free factor complexes I: Hyperbolicity, arXiv:1407.3508.
  • [HK] S. Hensel and D. Kielak, Nielsen realization for untwisted right-angled Artin groups, arXiv:1410.1618.
  • [HOP] S. Hensel, D. Osajda, and P. Przytycki, Realisation and dismantlability, Geom. Topol. 18 (2014), 2079–2126.
  • [Hor] C. Horbez, The boundary of the outer space of a free product, arXiv:1408.0543.
  • [KPS] A. Karrass, A. Pietrowski, and D. Solitar, Finite and infinite cyclic extensions of free groups, J. Aust. Math. Soc. 16 (1973), 458–466, Collection of articles dedicated to the memory of Hanna Neumann, IV.
  • [Ker1] S. P. Kerckhoff, The Nielsen realization problem, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 452–454.
  • [Ker2] S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), 235–265.
  • [Khr] D. G. Khramtsov, Finite groups of automorphisms of free groups, Mat. Zametki 38 (1985), 386–392, 476.
  • [Krö] B. Krön, Cutting up graphs revisited—A short proof of Stallings’ structure theorem, Groups Complex. Cryptol. 2 (2010), 213–221.
  • [Ser] J.-P. Serre, Trees, Springer Monogr. Math., Springer-Verlag, Berlin, 2003, translated from the French original by John Stillwell, corrected 2nd printing of the 1980 English translation.
  • [Sta1] J. R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334.
  • [Sta2] J. R. Stallings, Group theory and three-dimensional manifolds, Yale University Press, New Haven, CT, 1971.
  • [Zim] B. Zimmermann, Über Homöomorphismen $n$-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv. 56 (1981), 474–486.