2000 character limit reached
Relative hyperbolicity of free extensions of free groups (2307.09674v3)
Published 18 Jul 2023 in math.GR and math.GT
Abstract: We give necessary and sufficient conditions for a free-by-free group to be relatively hyperbolic with a cusp-preserving structure. Namely, if $\phi_1, \ldots , \phi_k $ is a collection of exponentially growing outer automorphisms with a common invariant \emph{subgroup system} such that any conjugacy class in the complement of this system grows exponentially under iteration by all $\phi_i$, then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of $\mathbb F$ by the free group generated by sufficiently high powers of $\phi_1, \ldots , \phi_k $, will be hyperbolic.
- M. Bestvina and M. Feighn. A combination theorem for negatively curved groups. J. Differential Geom., 35(1):85–101, 1992.
- Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal., 7(2):215–244, 1997.
- The Tits alternative for Out(Fn)Outsubscript𝐹𝑛{\rm Out}(F_{n})roman_Out ( italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). I. Dynamics of exponentially-growing automorphisms. Ann. of Math. (2), 151(2):517–623, 2000.
- Train tracks and automorphisms of free groups. Ann. of Math. (2), 135(1):1–51, 1992.
- B. H. Bowditch. Geometrical finiteness for hyperbolic groups. J. Funct. Anal., 113(2):245–317, 1993.
- B. H. Bowditch. Geometrical finiteness with variable negative curvature. Duke Math. J., 77(1):229–274, 1995.
- Brian H. Bowditch. The Cannon-Thurston map for punctured-surface groups. Math. Z., 255(1):35–76, 2007.
- B. H. Bowditch. Relatively hyperbolic groups. Internat. J. Algebra Comput., 22(3):1250016, 66, 2012.
- P. Brinkmann. Hyperbolic automorphisms of free groups. Geom. Funct. Anal., 10(5):1071–1089, 2000.
- Daryl Cooper. Automorphisms of free groups have finitely generated fixed point sets. J. Algebra, 111(2):453–456, 1987.
- François Dahmani. Les groupes relativement hyperboliques et leurs bords, volume 2003/13 of Prépublication de l’Institut de Recherche Mathématique Avancée [Prepublication of the Institute of Advanced Mathematical Research]. Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 2003. Thèse, l’Université Louis Pasteur (Strasbourg I), Strasbourg, 2003.
- François Dahmani and Suraj Krishna M S. Relative hyperbolicity of hyperbolic-by-cyclic groups. Groups Geom. Dyn., 17(2):403–426, 2023.
- Relative hyperbolicity for automorphisms of free products and free groups. J. Topol. Anal., 14(1):55–92, 2022.
- Hyperbolic extensions of free groups. Geom. Topol., 22(1):517–570, 2018.
- B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal., 8(5):810–840, 1998.
- The recognition theorem for Out(Fn)Outsubscript𝐹𝑛{\rm Out}(F_{n})roman_Out ( italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). Groups Geom. Dyn., 5(1):39–106, 2011.
- Pritam Ghosh. Limits of conjugacy classes under iterates of hyperbolic elements of Out(𝔽)Out𝔽{\rm Out}(\mathbb{F})roman_Out ( blackboard_F ). Groups Geom. Dyn., 14(1):177–211, 2020.
- Pritam Ghosh. Relative hyperbolicity of free-by-cyclic extensions. Compos. Math., 159(1):153–183, 2023.
- Regluing graphs of free groups. Algebr. Geom. Topol., 22(4):1969–2006, 2022.
- M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75–263. Springer, New York, 1987.
- Mark Hagen. A remark on thickness of free-by-cyclic groups. Illinois J. Math., 63(4):633–643, 2019.
- M. Handel and Lee Mosher. Subgroup classification in Out(Fnsubscript𝐹𝑛F_{n}italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT). arXiv, 2009.
- The free splitting complex of a free group, II: Loxodromic outer automorphisms. Trans. Amer. Math. Soc., 372(6):4053–4105, 2019.
- Subgroup decomposition in Out(Fn)Outsubscript𝐹𝑛{\rm Out}(F_{n})roman_Out ( italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). Mem. Amer. Math. Soc., 264(1280):vii+276, 2020.
- G. Christopher Hruska. Relative hyperbolicity and relative quasiconvexity for countable groups. Algebr. Geom. Topol., 10(3):1807–1856, 2010.
- Lee Mosher. A hyperbolic-by-hyperbolic hyperbolic group. Proc. Amer. Math. Soc., 125(12):3447–3455, 1997.
- A combination theorem for strong relative hyperbolicity. Geom. Topol., 12(3):1777–1798, 2008.
- Algebraic ending laminations and quasiconvexity. Algebr. Geom. Topol., 18(4):1883–1916, 2018.
- A combination theorem for metric bundles. Geom. Funct. Anal., 22(6):1636–1707, 2012.
- Denis V. Osin. Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Amer. Math. Soc., 179(843):vi+100, 2006.
- Abhijit Pal. Relatively hyperbolic extensions of groups and Cannon-Thurston maps. Proc. Indian Acad. Sci. Math. Sci., 120(1):57–68, 2010.
- William P. Thurston. Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle. Preprint, arXiv:math/9801045.
- Caglar Uyanik. Hyperbolic extensions of free groups from atoroidal ping-pong. Algebr. Geom. Topol., 19(3):1385–1411, 2019.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Collections
Sign up for free to add this paper to one or more collections.