No CrossRef data available.
Article contents
Relative hyperbolicity of free-by-cyclic extensions
Part of:
Structure and classification of infinite or finite groups
Low-dimensional topology
Special aspects of infinite or finite groups
Published online by Cambridge University Press: 30 January 2023
Abstract
Given a finitely generated free group 
$ {\mathbb {F} }$ of 
$\mathsf {rank}( {\mathbb {F} } )\geq 3$, we show that the mapping torus of 
$\phi$ is (strongly) relatively hyperbolic if 
$\phi$ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if 
$\phi$ has finite order.
Keywords
MSC classification
Primary:
20F65: Geometric group theory
- Type
- Research Article
- Information
- Copyright
- © 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
Footnotes
The author was supported by a faculty research grant from Ashoka University during this project.
References
Behrstock, J., Drutu, C. and Mosher, L., Thick metric spaces, relative hyperbolicity and quasi-isometric rigidity, Math. Ann. 344 (2009), 543–595.CrossRefGoogle Scholar
Bestvina, M. and Feighn, M., A combination theorem for negetively curved groups, J. Differential Geom. 43 (1992), 85–101.Google Scholar
Bestvina, M., Feighn, M. and Handel, M., Tits alternative in Out
$(F_n)$-I: Dynamics of exponentially-growing automorphisms, Ann. of Math. 151 (2000), 517–623.CrossRefGoogle Scholar
$(F_n)$-I: Dynamics of exponentially-growing automorphisms, Ann. of Math. 151 (2000), 517–623.CrossRefGoogle ScholarBestvina, M., Feighn, M. and Handel, M., The Tits alternative for Out$(F_n)$ II: A Kolchin type theorem, Ann. of Math. 161 (2005), 1–59.CrossRefGoogle Scholar
$(F_n)$ II: A Kolchin type theorem, Ann. of Math. 161 (2005), 1–59.CrossRefGoogle ScholarBestvina, M. and Handel, M., Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992), 1–51.CrossRefGoogle Scholar
Bestvina, M., Handel, M. and Feighn, M., Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), 215–244.CrossRefGoogle Scholar
Bowditch, B., The Cannon–Thurston map for punctured-surface groups, Math. Z. 255 (2007), 35–76.CrossRefGoogle Scholar
Bowditch, B., Relatively hyperolic groups, Internat. J. Algebra Comput. 22 (2012), 1250016.CrossRefGoogle Scholar
Bridson, M. and Groves, D., The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Mem. Amer. Math. Soc. 203 (2010), 1–152.Google Scholar
Brinkmann, P., Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000), 1071–1089.CrossRefGoogle Scholar
Button, J. and Kropholler, R., Nonhyperbolic free-by-cyclic and one-relator groups, New York J. Math. 22 (2016), 755–774.Google Scholar
Dowdall, S. and Taylor, S., Hyperbolic extensions of free groups, Geom. Topol. 22 (2018), 517–570.CrossRefGoogle Scholar
Drutu, C., Relatively hyperbolic groups: geometry and quasi-isometric invariance, Comment. Math. Helv. 84 (2009), 503–546.CrossRefGoogle Scholar
Feighn, M. and Handel, M., The recognition theorem for Out$(F_n)$, Groups Geom. Dyn. 5 (2011), 39–106.CrossRefGoogle Scholar
$(F_n)$, Groups Geom. Dyn. 5 (2011), 39–106.CrossRefGoogle ScholarGautero, F., Geodesics in trees of hyperbolic and relatively hyperbolic spaces, Proc. Edinb. Math. Soc. 59 (2016), 701–740.CrossRefGoogle Scholar
Gautero, F. and Lustig, M., The mapping torus group of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth, Preprint (2007), arXiv:0707.0822.Google Scholar
Genevois, A. and Horbez, C., Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups, J. Topol. 14 (2021), 963–991.CrossRefGoogle Scholar
Ghosh, P., Relatively irreducible free subgroups of Out $(\mathbb {F})$, Preprint (2018), arXiv:1210.8081.Google Scholar
$(\mathbb {F})$, Preprint (2018), arXiv:1210.8081.Google ScholarGhosh, P., Limits of conjugacy classes under iterates of hyperbolic elements of Out$(\mathbb {F})$, Groups Geom. Dyn. 4 (2021), 177–211.Google Scholar
$(\mathbb {F})$, Groups Geom. Dyn. 4 (2021), 177–211.Google ScholarHagen, M., A remark on thickness of free-by-cyclic groups, Illinois J. Math. 63 (2019), 633–643.CrossRefGoogle Scholar
Hagen, M. and Wise, D., Cubulating mapping tori of polynomial growth free group automorphisms, Preprint (2016), arXiv:1605.07879.Google Scholar
Handel, M. and Mosher, L., Subgroup classification in Out $(F_n)$, Preprint (2009), arXiv:0908.1255.Google Scholar
$(F_n)$, Preprint (2009), arXiv:0908.1255.Google ScholarHandel, M. and Mosher, L., Subgroup decomposition in Out$(F_n)$, Mem. Amer. Math. Soc. 264 (2020).Google Scholar
$(F_n)$, Mem. Amer. Math. Soc. 264 (2020).Google ScholarMacura, N., Quadratic isoperimetric inequality for mapping tori of polynomially growing outer automorphisms of free groups, Geom. Funct. Anal. 10 (2000), 874–901.CrossRefGoogle Scholar
Macura, N., Detour functions and quasi-isometries, Q. J. Math. 53 (2002), 207–239.CrossRefGoogle Scholar
Minasyan, A. and Osin, D., Acylindrical hyperbolicity of groups acting on trees, Math. Ann. 362 (2015), 1055–1105.CrossRefGoogle Scholar
Mj, M. and Reeves, L., A combination theorem for strong relative hyperolicity, Geom. Topol. 12 (2008), 1777–1798.CrossRefGoogle Scholar
Mosher, L., A hyperbolic-by-hyperbolic hyperbolic group, Proc. Amer. Math. Soc. 125 (1997), 3447–3455.CrossRefGoogle Scholar
Uyanik, C., Hyperbolic extensions of free groups using atoroidal ping-pong, Algebr. Geom. Topol. 19 (2019), 1385–1411.CrossRefGoogle Scholar