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Cubulating hyperbolic free-by-cyclic groups: the general case. (English) Zbl 1368.20050
Summary: Let \(\Phi\colon F\to F\) be an automorphism of the finite-rank free group \(F\). Suppose that \(G=F\rtimes_\Phi\mathbb Z\) is word-hyperbolic. Then \(G\) acts freely and cocompactly on a CAT(0) cube complex.

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E08 Groups acting on trees
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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[1] Agol, I., The virtual haken conjecture (with an appendix by ian agol, daniel groves and jason Manning), Documenta Mathematica Journal der DMV,, 18, 1045-1087, (2013) · Zbl 1286.57019
[2] M. Bestvina, M. Feighn and M. Handel. Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis (2)7 (1997), 215-244. · Zbl 0884.57002
[3] M. Bestvina, M. Feighn and M. Handel. The Tits alternative for Out(\(F\)_{\(n\)}) I: dynamics of exponentially-growing automorphisms. Annals of Mathematics 151 (2000), 517-623. · Zbl 0984.20025
[4] Bestvina, M.; Handel, M., Train tracks and automorphisms of free groups, Annals of Mathematics, 135, 1-51, (1992) · Zbl 0757.57004
[5] M.R. Bridson and A. Haefliger. Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999). · Zbl 0988.53001
[6] P. Brinkmann. Hyperbolic automorphisms of free groups. Geometric and Functional Analysis (5)10 (2000), 1071-1089. · Zbl 0970.20018
[7] N. Bergeron and D.T. Wise. A boundary criterion for cubulation. The American Journal of Mathematics (3)134 (2012), 843-859. doi:10.1353/ajm.2012.0020 · Zbl 1279.20051
[8] H. Bigdely and D.T. Wise. Quasiconvexity and relatively hyperbolic groups that split. Michigan Mathematical Journal (2)62 (2013), 387-406. · Zbl 1296.20041
[9] W. Dicks and E. Ventura. The Group Fixed by a Family of Injective Endomorphisms of a Free Group. Contemporary Mathematics. American Mathematical Society, Providence (1996). · Zbl 0845.20018
[10] Gersten, S.M., The automorphism group of a free group is not a CAT(0) group, Proceedings of the American Mathematical Society, 121, 999-1002, (1994) · Zbl 0807.20034
[11] F. Gautero and M. Lustig. The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth (2007), arXiv:0707.0822.
[12] M. Gromov. Hyperbolic groups. In: Essays in Group Theory, Math. Sci. Res. Inst. Publ., Vol. 8, pp. 75-263. Springer, New York (1987). · Zbl 0634.20015
[13] M.F. Hagen and P. Przytycki. Cocompactly cubulated graph manifolds. Israel Journal of Mathematics (2013), 1-11. arXiv:1311.2084. (To appear). · Zbl 1330.57030
[14] G.C. Hruska and D.T. Wise. Finiteness properties of cubulated groups. Compositio Mathematica (3)150 (2014), 453-506. doi:10.1112/S0010437X13007112. · Zbl 1335.20043
[15] F. Haglund and D.T. Wise. Special cube complexes. Geometric and Functional Analysis (5)17, 1551-1620. · Zbl 1155.53025
[16] T. Hsu and D.T. Wise. Cubulating malnormal amalgams. Inventiones mathematicae (199)2 (2015), 293-331. · Zbl 1320.20039
[17] M.F. Hagen and D.T. Wise. Cubulating hyperbolic free-by-cyclic groups: the irreducible case (2013), 1-39. arXiv:1311.2084. (Preprint)
[18] G. Levitt. Counting growth types of automorphisms of free groups. Geometric and Functional Analysis (4)19 (2009), 1119-1146. · Zbl 1196.20038
[19] M. Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society (3)71 (1995), 585-617. · Zbl 0861.20041
[20] D.T. Wise. Cubular tubular groups. Transactions of the American Mathematical Society 366 (2014), 5503-5521. doi:10.1090/S0002-9947-2014-06065-0. · Zbl 1346.20058
[21] D.T. Wise. The structure of groups with a quasiconvex hierarchy (2011), 205. (Preprint)
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