Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings. (English) Zbl 1361.20029
Let \(G_1,\ldots ,G_k\) be nontrivial countable groups and \(F_N\) the free group of rank \(N\). Define \(G=G_1*\cdots *G_k*F_N\). Denote by \({\mathcal F}=\{ [ G_1 ],\ldots ,[ G_k ]\}\) the finite collection of \(G\)-conjugacy classes of the subgroups \(G_1,\ldots ,G_k\) which is called a free factor system of \(G\).
The author defines analogues of the graphs of free splittings, of cyclic splittings and of maximally cyclic splittings of \(F_N\) for such a free factor system. The Gromov boundary of the graph of relative cyclic splittings is identified with the space of equivalence classes of \({\mathcal Z}\)-averse trees in the boundary of the corresponding outer space. A tree is \({\mathcal Z}\)-averse if it is not compatible with any tree that is itself compatible with a relative cyclic splitting.
A similar description is given of the Gromov boundary of the graph of maximally cyclic splittings.
The author defines analogues of the graphs of free splittings, of cyclic splittings and of maximally cyclic splittings of \(F_N\) for such a free factor system. The Gromov boundary of the graph of relative cyclic splittings is identified with the space of equivalence classes of \({\mathcal Z}\)-averse trees in the boundary of the corresponding outer space. A tree is \({\mathcal Z}\)-averse if it is not compatible with any tree that is itself compatible with a relative cyclic splitting.
A similar description is given of the Gromov boundary of the graph of maximally cyclic splittings.
Reviewer: Stephan Rosebrock (Karlsruhe)
MSC:
| 20F65 | Geometric group theory |
| 57M07 | Topological methods in group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20F69 | Asymptotic properties of groups |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
References:
| [1] | Y.Algom‐Kfir, ‘The metric completion of outer space’, Preprint, 2013, arXiv:1202.6392v4. |
| [2] | G. C.Bell, K.Fujiwara, ‘The asymptotic dimension of a curve graph is finite’, J. London Math. Soc., 77 (2008) 33-50. · Zbl 1135.57010 |
| [3] | M.Bestvina, K.Bromberg, K.Fujiwara, ‘Constructing group actions on quasi‐trees and applications to mapping class groups’, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1-64. · Zbl 1372.20029 |
| [4] | M.Bestvina, M.Feighn, ‘Stable actions of groups on real trees’, Invent. Math., 121 (1995) 287-321. · Zbl 0837.20047 |
| [5] | M.Bestvina, M.Feighn, ‘Hyperbolicity of the complex of free factors’, Adv. Math., 256 (2014) 104-155. · Zbl 1348.20028 |
| [6] | M.Bestvina, M.Feighn, ‘Subfactor projections’, J. Topol., 7 (2014) 771-804. · Zbl 1346.20054 |
| [7] | M.Bestvina, M.Feighn, M.Handel, ‘Laminations, trees, and irreducible automorphisms of free groups’, Geom. Funct. Anal., 7 (1997) 215-244. · Zbl 0884.57002 |
| [8] | M.Bestvina, K.Fujiwara, ‘Bounded cohomology of subgroups of mapping class groups’, Geom. Topol., 6 (2002) 69-89. · Zbl 1021.57001 |
| [9] | M.Bestvina, P.Reynolds, ‘The boundary of the complex of free factors’, Duke Math. J. 164 (2015) 2213-2251. · Zbl 1337.20040 |
| [10] | M. R.Bridson, A.Haefliger, Metric spaces of non‐positive curvature (Springer, Berlin, 1999). · Zbl 0988.53001 |
| [11] | J. F.Brock, R. D.Canary, Y. N.Minsky, ‘The classification of Kleinian surface groups, II: the ending lamination conjecture’, Ann. Math., 176 (2012) 1-149. · Zbl 1253.57009 |
| [12] | M.Coornaert, T.Delzant, A.Papadopoulos, Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441 (Springer, Berlin, 1990). · Zbl 0727.20018 |
| [13] | M.Culler, J. W.Morgan, ‘Group actions on \(\mathbb{R} \)‐trees’, Proc. London Math. Soc., 55 (1987) 571-604. · Zbl 0658.20021 |
| [14] | B.Farb, L.Mosher, ‘Convex cocompact subgroups of mapping class groups’, Geom. Topol., 91-152 (2002) 91-152. · Zbl 1021.20034 |
| [15] | S.Francaviglia, A.Martino, ‘Stretching factors, metrics and train tracks for free products’, Preprint, 2015, arXiv:1312.4172v3. |
| [16] | D.Gabai, ‘Almost filling laminations and the connectivity of ending lamination space’, Geom. Topol., 13 (2009) 1017-1041. · Zbl 1165.57015 |
| [17] | E.Ghys, P.de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov (Birkhäuser, Basel, 1990). · Zbl 0731.20025 |
| [18] | M.Gromov, ‘Hyperbolic groups’, Essays in group theory, MSRI Publications 8 (ed. S. M. Gersten; Springer, Berlin, 1987) 75-265. · Zbl 0634.20015 |
| [19] | V.Guirardel, ‘Approximations of stable actions on \(\mathbb{R} \)‐trees’, Comment. Math. Helv., 73 (1998) 89-121. · Zbl 0979.20026 |
| [20] | V.Guirardel, ‘Limit groups and groups acting freely on \(\mathbb{R}^n\)‐trees’, Geom. Topol., 8 (2004) 1427-1470. · Zbl 1114.20013 |
| [21] | V.Guirardel, ‘Actions of finitely generated groups on \(\mathbb{R} \)‐trees’, Ann. Inst. Fourier, 58 (2008) 159-211. · Zbl 1187.20020 |
| [22] | V.Guirardel, G.Levitt, ‘Deformation spaces of trees’, Groups Geom. Dyn., 1 (2007) 135-181. · Zbl 1134.20026 |
| [23] | V.Guirardel, G.Levitt, ‘The outer space of a free product’, Proc. London Math. Soc., 94 (2007) 695-714. · Zbl 1168.20011 |
| [24] | V.Guirardel, G.Levitt, ‘JSJ decompositions: definitions, existence, uniqueness. II: compatibility and acylindricity’, Preprint, 2010, arXiv:1002.4564v2. |
| [25] | V.Guirardel, G.Levitt, ‘Splittings and automorphisms of relatively hyperbolic groups’, Groups. Geom. Dyn., 9 (2015) 599-663. · Zbl 1368.20037 |
| [26] | U.Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, London Mathematical Society Lecture Note Series, 329 (Cambridge University Press, Cambridge, 2006) 187-207. · Zbl 1117.30036 |
| [27] | U.Hamenstädt, ‘Geometry of the mapping class group I: boundary amenability’, Invent. Math., 175 (2009) 545-609. · Zbl 1197.57003 |
| [28] | U.Hamenstädt, ‘The boundary of the free splitting graph and of the free factor graph’, Preprint, 2014, arXiv:1211.1630v5. |
| [29] | U.Hamenstädt, S.Hensel, ‘Spheres and projections for \(\text{Out} ( F_n )\)’, J. Topol. 8 (2015) 65-92. · Zbl 1367.20046 |
| [30] | M.Handel, L.Mosher, ‘The free splitting complex of a free group I: hyperbolicity’, Geom. Topol., 17 (2013) 1581-1670. · Zbl 1278.20053 |
| [31] | A.Hilion, C.Horbez, ‘The hyperbolicity of the sphere complex via surgery paths’, J. reine angew. Math. (2015), to appear, doi 10.1515/crelle‐2014‐0128. · Zbl 1379.57004 |
| [32] | C.Horbez, ‘Spectral rigidity for primitive elements of \(F_N\)’, J. Group Theory, to appear. · Zbl 1350.20022 |
| [33] | C.Horbez, ‘The boundary of the outer space of a free product’, Preprint, 2014, arXiv:1408.0543v2. |
| [34] | C.Horbez, ‘The Tits alternative for the automorphism group of a free product’, Preprint, 2014, arXiv:1408.0546v2. |
| [35] | I.Kapovich, K.Rafi, ‘On hyperbolicity of free splitting and free factor complexes’, Groups Geom. Dyn., 8 (2014) 391-414. · Zbl 1315.20022 |
| [36] | Y.Kida, ‘The mapping class group from the viewpoint of measure equivalence theory’, Mem. Amer. Math. Soc., 196 (2008). · Zbl 1277.37008 |
| [37] | E.Klarreich, ‘The boundary at infinity of the curve complex and the relative Teichmüller space’, Preprint, 1999, https://pressfolios‐production.s3.amazonaws.com/uploads/story/story_pdf/145710/1457101434403642.pdf. |
| [38] | T.Kobayashi, ‘Heights of simple loops and pseudo‐Anosov homeomorphisms’, Contemp. Math., 78 (1988) 327-338. · Zbl 0663.57010 |
| [39] | A. G.Kurosh, ‘Die Untergruppen der freien Produkte von beliebigen Gruppen’, Math. Ann., 109 (1934) 647-660. · Zbl 0009.01004 |
| [40] | R. C.Lacher, ‘Cell‐like mappings, I’, Pacific J. Math., 30 (1969) 717-731. · Zbl 0182.57601 |
| [41] | C. J.Leininger, S.Schleimer, ‘Connectivity of the space of ending laminations’, Duke Math. J., 150 (2009) 533-575. · Zbl 1190.57013 |
| [42] | G.Levitt, ‘Graphs of actions on \(\mathbb{R} \)‐trees’, Comment. Math. Helv., 69 (1994) 28-38. · Zbl 0802.05044 |
| [43] | B.Mann, ‘Hyperbolicity of the cyclic splitting graph’, Geom. Dedicata, 173 (2014) 271-280. · Zbl 1336.20043 |
| [44] | H. A.Masur, Y. N.Minsky, ‘Geometry of the complex of curves I: hyperbolicity’, Invent. Math., 138 (1999) 103-149. · Zbl 0941.32012 |
| [45] | H. A.Masur, S.Schleimer, ‘The geometry of the disk complex’, J. Amer. Math. Soc., 26 (2013) 1-62. · Zbl 1272.57015 |
| [46] | Y. N.Minsky, ‘The classification of Kleinian surface groups I: models and bounds’, Ann. Math., 171 (2010) 1-107. · Zbl 1193.30063 |
| [47] | J. W.Morgan, ‘Ergodic theory and free actions of groups on \(\mathbb{R} \)‐trees’, Invent. Math., 94 (1988) 605-622. · Zbl 0676.57001 |
| [48] | F.Paulin, ‘Topologie de Gromov équivariante, structures hyperboliques et arbres réels’, Invent. Math., 94 (1988) 53-80. · Zbl 0673.57034 |
| [49] | K.Rafi, S.Schleimer, ‘Curve complexes are rigid’, Duke Math. J., 158 (2011) 225-246. · Zbl 1227.57024 |
| [50] | P.Reynolds, ‘Dynamics of irreducible endomorphisms of \(F_n\)’, Preprint, 2011, arXiv:1008.3659v3. |
| [51] | P.Reynolds, ‘Reducing systems for very small trees’, Preprint, 2012, arXiv:1211.3378v1. |
| [52] | P.Scott, T.Wall, ‘Topological methods in group theory’, Homological group theory (ed. C. T. C. Wall), LMS Lecture Notes Series 36 (Cambridge University Press, Cambridge, 1979), 137-203. · Zbl 0423.20023 |
| [53] | J. J.Walsh, ‘Dimension, cohomological dimension, and cell‐like mappings’, Shape theory and geometric topology, Lecture Notes in Mathematics 870 (Springer, Berlin, 1981) 105-118. · Zbl 0474.55002 |
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