Hyperbolicity of the complex of free factors.

*(English)*Zbl 1348.20028
Adv. Math. 256, 104-155 (2014); corrigendum ibid. 259, 843 (2014).

From the introduction: We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.

The complex of free factors of a free group \(\mathbb F\) of rank \(n\) is the simplicial complex \(\mathcal F\) whose vertices are conjugacy classes of proper free factors \(A\) of \(\mathbb F\), and simplices are determined by chains \(A_1<A_2<\cdots<A_k\).…There is a very useful analogy between \(\mathcal F\) and the curve complex \(\mathcal C\) associated with a compact surface (with punctures) \(\Sigma\). The vertices of \(\mathcal C\) are isotopy classes of essential simple closed curves in \(\Sigma\), and simplices are determined by pairwise disjoint curves.

More recently, the curve complex has been used in the study of the geometry of mapping class groups and of ends of hyperbolic 3-manifolds. The fundamental result on which this work is based is the theorem of H. A. Masur and Y. N. Minsky [Invent. Math. 138, No. 1, 103-149 (1999; Zbl 0941.32012)] that the curve complex is hyperbolic. In the low complexity cases when \(\mathcal C\) is a discrete set one modifies the definition of \(\mathcal C\) by adding an edge when the two curves intersect minimally. In the same way, we modify the definition of \(\mathcal F\) when the rank is \(n=2\) by adding an edge when the two free factors (necessarily of rank 1) are determined by a basis of \(\mathbb F\), i.e. whenever \(\mathbb F=\langle a,b\rangle\), then \(\langle a\rangle\) and \(\langle b\rangle\) span an edge. In this way \(\mathcal F\) becomes the standard Farey graph. The main result in this paper is:

Main Theorem. The complex \(\mathcal F\) of free factors is hyperbolic.

The statement simply means that when the 1-skeleton of \(\mathcal F\) is equipped with the path metric in which every edge has length 1, the resulting graph is hyperbolic.

The complex of free factors of a free group \(\mathbb F\) of rank \(n\) is the simplicial complex \(\mathcal F\) whose vertices are conjugacy classes of proper free factors \(A\) of \(\mathbb F\), and simplices are determined by chains \(A_1<A_2<\cdots<A_k\).…There is a very useful analogy between \(\mathcal F\) and the curve complex \(\mathcal C\) associated with a compact surface (with punctures) \(\Sigma\). The vertices of \(\mathcal C\) are isotopy classes of essential simple closed curves in \(\Sigma\), and simplices are determined by pairwise disjoint curves.

More recently, the curve complex has been used in the study of the geometry of mapping class groups and of ends of hyperbolic 3-manifolds. The fundamental result on which this work is based is the theorem of H. A. Masur and Y. N. Minsky [Invent. Math. 138, No. 1, 103-149 (1999; Zbl 0941.32012)] that the curve complex is hyperbolic. In the low complexity cases when \(\mathcal C\) is a discrete set one modifies the definition of \(\mathcal C\) by adding an edge when the two curves intersect minimally. In the same way, we modify the definition of \(\mathcal F\) when the rank is \(n=2\) by adding an edge when the two free factors (necessarily of rank 1) are determined by a basis of \(\mathbb F\), i.e. whenever \(\mathbb F=\langle a,b\rangle\), then \(\langle a\rangle\) and \(\langle b\rangle\) span an edge. In this way \(\mathcal F\) becomes the standard Farey graph. The main result in this paper is:

Main Theorem. The complex \(\mathcal F\) of free factors is hyperbolic.

The statement simply means that when the 1-skeleton of \(\mathcal F\) is equipped with the path metric in which every edge has length 1, the resulting graph is hyperbolic.

##### MSC:

20E05 | Free nonabelian groups |

20F65 | Geometric group theory |

57M50 | General geometric structures on low-dimensional manifolds |

##### References:

[1] | Algom-Kfir, Y., Strongly contracting geodesics in outer space, Geom. Topol., 15, 4, 2181-2233, (2011) |

[2] | Algom-Kfir, Y.; Bestvina, M., Asymmetry of outer space, Geom. Dedicata, 156, 81-92, (2012) |

[3] | Bestvina, M.; Feighn, M.; Handel, M., Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal., 7, 2, 215-244, (1997) |

[4] | Bestvina, M., A bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math., 214, 1, 1-12, (2011) |

[5] | Bestvina, M.; Feighn, M., A hyperbolic \(\operatorname{Out}(F_n)\)-complex, Groups Geom. Dyn., 4, 1, 31-58, (2010) |

[6] | Bestvina, M.; Fujiwara, K., Bounded cohomology of subgroups of mapping class groups, Geom. Topol., 6, 69-89, (2002), (electronic) |

[7] | Bowditch, B. H., Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math., 598, 105-129, (2006) |

[8] | Clay, M., Contractibility of deformation spaces of G-trees, Algebr. Geom. Topol., 5, 1481-1503, (2005), (electronic) |

[9] | Culler, M.; Vogtmann, K., Moduli of graphs and automorphisms of free groups, Invent. Math., 84, 1, 91-119, (1986) |

[10] | Day, M.; Putman, A., The complex of partial bases for \(F_n\) and finite generation of the Torelli subgroup of \(A u t(F_n)\), Geom. Dedicata, 164, 139-153, (2013) |

[11] | Francaviglia, S.; Martino, A., Metric properties of outer space, Publ. Mat., 55, 433-473, (2011) |

[12] | Guirardel, V.; Levitt, G., Deformation spaces of trees, Groups Geom. Dyn., 1, 2, 135-181, (2007) |

[13] | Handel, M.; Mosher, L., Subgroup classification in \(O u t(F_n)\) |

[14] | Handel, M.; Mosher, L., The free splitting complex of a free group, I: hyperbolicity, Geom. Topol., 17, 3, 1581-1672, (2013) |

[15] | Harer, J. L., Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2), 121, 2, 215-249, (1985) |

[16] | Harer, J. L., The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., 84, 1, 157-176, (1986) |

[17] | Harvey, W. J., Boundary structure of the modular group, (Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, NY, 1978, Ann. of Math. Stud., vol. 97, (1981), Princeton Univ. Press Princeton, NJ), 245-251 |

[18] | Hatcher, A.; Vogtmann, K.; Wahl, N., Erratum to: “homology stability for outer automorphism groups of free groups” [algebr. geom. topol. 4 (2004) 1253-1272], Algebr. Geom. Topol., 6, 573-579, (2006), (electronic) |

[19] | Hatcher, A.; Vogtmann, K., Cerf theory for graphs, J. Lond. Math. Soc. (2), 58, 3, 633-655, (1998) |

[20] | Hatcher, A.; Vogtmann, K., The complex of free factors of a free group, Q. J. Math. Oxford Ser. (2), 49, 196, 459-468, (1998) |

[21] | Hatcher, A.; Vogtmann, K., Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol., 4, 1253-1272, (2004) |

[22] | Hilion, A.; Horbez, C., The hyperbolicity of the sphere complex via surgery paths |

[23] | Ilya, K.; Lustig, M., Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol., 13, 3, 1805-1833, (2009) |

[24] | Ilya, K.; Rafi, K., On hyperbolicity of free splitting and free factor complexes, (2014), Groups, Geometry, and Dynamics., in press |

[25] | Mann, B., Hyperbolicity of the cyclic splitting complex, Geom. Dedicata, (2014), in press |

[26] | Reiner, M., Non-uniquely ergodic foliations of thin type, measured currents, and automorphisms of free groups, (1995), UCLA, PhD thesis |

[27] | Masur, H. A.; Minsky, Y. N., Geometry of the complex of curves. I. hyperbolicity, Invent. Math., 138, 1, 103-149, (1999) |

[28] | Minsky, Y. N., Quasi-projections in Teichmüller space, J. Reine Angew. Math., 473, 121-136, (1996) |

[29] | R.K. Skora, Deformations of length functions in groups, preprint, 1989. |

[30] | Stallings, J. R., Topology of finite graphs, Invent. Math., 71, 3, 551-565, (1983) |

[31] | Stallings, J. R., Whitehead graphs on handlebodies, (Geometric Group Theory Down Under, Canberra, 1996, (1999), de Gruyter Berlin), 317-330 |

[32] | Whitehead, J. H.C., On certain sets of elements in a free group, Proc. Lond. Math. Soc., 41, 48-56, (1936) |

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