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A combination theorem for strong relative hyperbolicity. (English) Zbl 1192.20027
From the introduction: We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn’s Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.
In [J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)], M. Bestvina and M. Feighn proved a combination theorem for hyperbolic groups. Motivated by this, Swarup asked the analogous question [in Geometric group theory problem list, available at http://math.utah.edu/~bestvina], for relatively hyperbolic groups. F. Dahmani [Geom. Topol. 7, 933-963 (2003; Zbl 1037.20042)] and E. Alibegović [Bull. Lond. Math. Soc. 37, No. 3, 459-466 (2005; Zbl 1074.57001)] have proven combination theorems motivated by applications to convergence groups and limit groups.
In this paper, we prove a geometric combination theorem (as opposed to a dynamical one) for trees of (strong) relatively hyperbolic metric spaces. We use Bestvina and Feighn’s Combination Theorem [op. cit.] directly in deducing the relevant combination theorem. The conditions we impose are quite different from those of Dahmani [op. cit.] and Alibegović [op. cit.]. Our main Theorems 4.5 and 4.7 are stated below:
Strong Combination Theorem and converse: Theorem 4.5 and Theorem 4.7: Let $$X$$ be a tree $$(T)$$ of strongly relatively hyperbolic spaces satisfying: (1) the q(uasi)-i(sometrically)-embedded condition, (2) the strictly type-preserving condition, (3) the qi-preserving electrocution condition, (4) the induced tree of coned-off spaces satisfies the hallways flare condition, (5) the cone-bounded hallways strictly flare condition. Then $$X$$ is strongly hyperbolic relative to the family $$\mathcal C$$ of maximal cone-subtrees of horosphere-like spaces.
Conversely, if $$X$$ is a tree $$(T)$$ of strongly relatively hyperbolic spaces satisfying conditions (1)-(3) such that $$X$$ is strongly hyperbolic relative to the family $$\mathcal C$$ of maximal cone-subtrees of horosphere-like spaces, then the tree of spaces satisfies conditions (4)-(5).
Reviewer: Reviewer (Berlin)

MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 57M50 General geometric structures on low-dimensional manifolds 20F65 Geometric group theory 20M07 Varieties and pseudovarieties of semigroups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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References:
 [1] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459 · Zbl 1074.57001 · doi:10.1112/S0024609304004059 [2] J Behrstock, C Drutu, L Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity · Zbl 1220.20037 · doi:10.1007/s00208-008-0317-1 [3] M Bestvina, Geometric group theory problem list, available at http://math.utah.edu/ bestvina · Zbl 0998.57003 [4] M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85 · Zbl 0724.57029 [5] M Bestvina, M Feighn, M Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997) 215 · Zbl 0884.57002 · doi:10.1007/PL00001618 [6] B H Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997) · Zbl 1259.20052 [7] B H Bowditch, The Cannon-Thurston map for punctured-surface groups, Math. Z. 255 (2007) 35 · Zbl 1138.57020 · doi:10.1007/s00209-006-0012-4 [8] I Bumagin, On definitions of relatively hyperbolic groups, Contemp. Math. 372, Amer. Math. Soc. (2005) 189 · Zbl 1091.20029 [9] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 · Zbl 1037.20042 · doi:10.2140/gt.2003.7.933 · eudml:123509 · arxiv:math/0203258 [10] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 · doi:10.1007/s000390050075 [11] F. Gautero, Geodesics in trees of hyperbolic and relatively hyperbolic groups · Zbl 1397.20060 · arxiv:0710.4079 [12] É Ghys, P d l Harpe, editor, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math. 83, Birkhäuser (1990) · Zbl 0731.20025 [13] R Gitik, M Mitra, E Rips, M Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998) 321 · Zbl 0897.20030 · doi:10.1090/S0002-9947-98-01792-9 [14] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [15] G Hruska, D Wise, Packing subgroups in relatively hyperbolic groups · Zbl 1188.20042 · doi:10.2140/gt.2009.13.1945 · arxiv:math/0609369 [16] M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math. 183, Birkhäuser (2001) · Zbl 0958.57001 [17] E Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere, Amer. J. Math. 121 (1999) 1031 · Zbl 1011.30035 · doi:10.1353/ajm.1999.0034 · muse.jhu.edu [18] M Mitra, Maps on boundaries of hyperbolic metric spaces, PhD thesis, UC Berkeley (1997) [19] M Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998) 135 · Zbl 0906.20023 [20] M Mitra, Height in splittings of hyperbolic groups, Proc. Indian Acad. Sci. Math. Sci. 114 (2004) 39 · Zbl 1059.20040 · doi:10.1007/BF02829670 [21] M Mj, Cannon-Thurston maps for pared manifolds of bounded geometry · Zbl 1166.57009 · doi:10.2140/gt.2009.13.189 · arxiv:math/0503581 [22] M Mj, Cannon-Thurston maps for surface groups I: amalgamation geometry and split geometry · Zbl 1301.57013 · doi:10.4007/annals.2014.179.1.1 [23] M Mj, Cannon-Thurston maps, i-bounded geometry and a theorem of McMullen · Zbl 1237.57018 · arxiv:math.GT/0511041 · eudml:116466 [24] M Mj, A Pal, Relative hyperbolicity, trees of spaces and Cannon-Thurston maps · Zbl 1222.57013 · doi:10.1007/s10711-010-9519-2 · arxiv:0708.3578 [25] L Mosher, Hyperbolic extensions of groups, J. Pure Appl. Algebra 110 (1996) 305 · Zbl 0851.20037 · doi:10.1016/0022-4049(95)00081-X [26] L Mosher, A hyperbolic-by-hyperbolic hyperbolic group, Proc. Amer. Math. Soc. 125 (1997) 3447 · Zbl 0895.20028 · doi:10.1090/S0002-9939-97-04249-4 [27] A Pal, Cannon-Thurston maps and relative hyperbolicity, PhD thesis, Indian Statistical Institute, Calcutta (expected 2009) [28] Z Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31 · Zbl 1018.20034 · doi:10.1007/s10240-001-8188-y · numdam:PMIHES_2001__93__31_0 · eudml:104176 [29] G A Swarup, Proof of a weak hyperbolization theorem, Q. J. Math. 51 (2000) 529 · Zbl 0965.57012 · doi:10.1093/qjmath/51.4.529
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