词条 | Free factor complex |
释义 |
In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Hatcher and Vogtmann.[1] Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometry of . Formal definitionFor a free group a proper free factor of is a subgroup such that and that there exists a subgroup such that . Let be an integer and let be the free group of rank . The free factor complex for is a simplicial complex where: (1) The 0-cells are the conjugacy classes in of proper free factors of , that is (2) For , a -simplex in is a collection of distinct 0-cells such that there exist free factors of such that for , and that . [The assumption that these 0-cells are distinct implies that for ]. In particular, a 1-cell is a collection of two distinct 0-cells where are proper free factors of such that . For the above definition produces a complex with no -cells of dimension . Therefore, is defined slightly differently. One still defines to be the set of conjugacy classes of proper free factors of ; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices determine a 1-simplex in if and only if there exists a free basis of such that . The complex has no -cells of dimension . For the 1-skeleton is called the free factor graph for . Main properties
Other modelsThere are several other models which produce graphs coarsely -equivariantly quasi-isometric to . These models include:
References1. ^1 Allen Hatcher, and Karen Vogtmann, The complex of free factors of a free group. Quarterly Journal of Mathematics, Oxford Ser. (2) 49 (1998), no. 196, pp. 459–468 2. ^Ilya Kapovich and Martin Lustig, Geometric intersection number and analogues of the curve complex for free groups. Geometry & Topology 13 (2009), no. 3, pp. 1805–1833 3. ^Jason Behrstock, Mladen Bestvina, and Matt Clay, Growth of intersection numbers for free group automorphisms. Journal of Topology 3 (2010), no. 2, pp. 280–310 4. ^1 2 Mladen Bestvina and Mark Feighn, Hyperbolicity of the complex of free factors. Advances in Mathematics 256 (2014), pp. 104–155 5. ^1 Ilya Kapovich and Kasra Rafi, On hyperbolicity of free splitting and free factor complexes. Groups, Geometry, and Dynamics 8 (2014), no. 2, pp. 391–414 6. ^Arnaud Hilion and Camille Horbez, [https://www.degruyter.com/view/j/crll.ahead-of-print/crelle-2014-0128/crelle-2014-0128.xml?format=INT The hyperbolicity of the sphere complex via surgery paths], Journal für die reine und angewandte Mathematik 730 (2017), 135–161 7. ^Michael Handel and Lee Mosher, The free splitting complex of a free group, I: hyperbolicity. Geometry & Topology, 17 (2013), no. 3, 1581--1672. {{MR|3073931}}{{doi|10.2140/gt.2013.17.1581}} 8. ^Mladen Bestvina and Patrick Reynolds, The boundary of the complex of free factors. Duke Mathematical Journal 164 (2015), no. 11, pp. 2213–2251 9. ^Camille Horbez, The Poisson boundary of . Duke Mathematical Journal 165 (2016), no. 2, pp. 341–369 See also
2 : Geometric group theory|Geometric topology |
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