Hostname: page-component-7bb8b95d7b-2h6rp Total loading time: 0 Render date: 2024-09-20T16:36:26.215Z Has data issue: true hasContentIssue false
  • English
  • Français

Centralisers of linear growth automorphisms of free groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  20 September 2024

NAOMI ANDREW  and
ARMANDO MARTINO
NAOMI ANDREW
Affiliation:
Mathematical Sciences, Building 54, University of Southampton, Southampton, SO17 1BJ. e-mails: A.Martino@soton.ac.uk, naomi.maths@gmail.com
ARMANDO MARTINO
Affiliation:
Mathematical Sciences, Building 54, University of Southampton, Southampton, SO17 1BJ. e-mails: A.Martino@soton.ac.uk, naomi.maths@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

MSC classification

Primary: 20E05: Free nonabelian groups 20E08: Groups acting on trees 20E36: Automorphisms of infinite groups 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Secondary: 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 22
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alperin, R. and Bass, H.. Length functions of group actions on $\Lambda$ -trees. In Combinatorial group theory and topology (Alta, Utah, 1984) 1987. Ann. of Math. Stud. 111 (Princeton University Press, Princeton, NJ), pp. 265–378.Google Scholar
Andrew, N.. Serre’s property (FA) for automorphism groups of free products. J. Group Theory 24 (2021), 385–414.Google Scholar
Andrew, N. and Martino, A.. Free-by-cyclic groups, automorphisms and actions on nearly canonical trees. J. Algebra 604 (2022), 451–495.Google Scholar
Bass, H.. Covering theory for graphs of groups. J. Pure Appl. Algebra 89 (1993), 3–47.Google Scholar
Bass, H. and Jiang, R.. Automorphism groups of tree actions and of graphs of groups. J. Pure Appl. Algebra 112 (1996), 109–155.Google Scholar
Bestvina, M., Feighn, M. and Handel, M.. Laminations trees, and irreducible automorphisms of free groups. Geom. Funct. Anal. 7 (1997), 215–244.Google Scholar
Bestvina, M., Feighn, M. and Handel, M.. The Tits alternative for $\text{Out}(F_n)$ . I: Dynamics of exponentially-growing automorphisms. Ann. of Math (2) 151 (2000), 517–623.Google Scholar
Bestvina, M., Feighn, M. and Handel, M.. A McCool Whitehead type theorem for finitely generated subgroups of $\mathrm{Out}(F_n)$ . Ann. H. Lebesgue 6 (2023), 65–94.Google Scholar
Bestvina, M. and Handel, M.. Train tracks and automorphisms of free groups. Ann. of Math (2) 135 (1992), 1–51.Google Scholar
Bridson, M. R.. Semisimple actions of mapping class groups on ${\rm CAT}(0)$ spaces. In Geometry of Riemann surfaces 2010. London Math. Soc. Lecture Note Ser., 368 (Cambridge University Press, Cambridge), pp. 1–14.Google Scholar
Cohen, M. M. and Lustig, M.. The conjugacy problem for Dehn twist automorphisms of free groups. Comment. Math. Helv. 74 (1999), 179–200.Google Scholar
Culler, M. and Morgan, J. W.. Group actions on $\mathbb{R}$ -trees. Proc. London Math. Soc. (3) 55 (1987), 571–604.Google Scholar
Culler, M. and Vogtmann, K.. Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), 91–119.Google Scholar
Dahmani, F.. On suspensions and conjugacy of hyperbolic automorphisms. Trans. Amer. Math. Soc. 368 (2016), 5565–5577.Google Scholar
Guerch, Y.. 2022. Roots of outer automorphisms of free groups and centralizers of abelian subgroups of $\mathrm{Out}(F_N)$ . ArXiv:2212.07674 Google Scholar
Guerch, Y., Hughes, S. and Sánchez Saldaña, L. J.. Centralisers and the virtually cyclic dimension of $\mathrm{Out}(F_N)$ . ArXiv:2308.01590 (2023).Google Scholar
Guirardel, V. and Levitt, G.. McCool groups of toral relatively hyperbolic groups. Algebr. Geom. Topol. 15 (2015), 3485–3534.Google Scholar
Ivanov, N. V.. Subgroups of Teichmüller modular groups. Trans. Math. Monogr. 115. (American Mathematical Society, Providence, RI, 1992). Translated from the Russian by E. J. F. Primrose and revised by the author.Google Scholar
Krsticć, S., Lustig, M. and Vogtmann, K.. An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms. Proc. Edinburgh Math. Soc. (2) 44 (2001), 117–141.Google Scholar
Krsticć, S. and Vogtmann, K.. Equivariant outer space and automorphisms of free-by-finite groups. Comment. Math. Helv. 68 (1993), 216–262.Google Scholar
Levitt, G.. Counting growth types of automorphisms of free groups. Geom. Funct. Anal. 19 (2009), 1119–1146.Google Scholar
Levitt, G.. Generalized Baumslag–Solitar groups: rank and finite index subgroups. Ann. Inst. Fourier (Grenoble) 65 (2015), 725–762.Google Scholar
Rafi, K., Selinger, N. and Yampolsky, M.. Centralizers in mapping class groups and decidability of Thurston equivalence. Arnold Math. J. 6 (2020), 271–290.Google Scholar
Robinson, D. J. S.. A course in the theory of groups. Grad. Texts in Math. 80 (Springer-Verlag, New York, 1996), second edition.Google Scholar
Rodenhausen, M. and Wade, R. D.. Centralisers of Dehn twist automorphisms of free groups. Math. Proc. Camb. Phil. Soc. 159 (2015), 89–114.Google Scholar
Serre, J.-P.. Trees. Springer Monogr. Math. (Springer-Verlag, Berlin, 2003). Translated from the French original by John Stillwell, corrected 2nd printing of the 1980 English translation.Google Scholar