×

The symmetries of outer space. (English) Zbl 1037.20025

The authors investigate the natural map \(\text{Out}(F_n)\to\operatorname{Aut}(K_n)\) from the outer automorphism group of the free group of rank \(n\), \(F_n\), to the group of the spine, \(K_n\), of outer space and prove that for \(n\geq 3\), this natural map is an isomorphism. The relationship between \(\text{Out}(F_n)\) and \(\operatorname{Aut}(K_n)\) is described as analogous to Royden’s Theorem which states that the full isometry group of the Teichmüller space associated to a compact surface of genus at least two is the mapping class group of the surface. Another analogy is made utilizing Tits’ Theorem which states that for surfaces of genus at least two, the full group of simplicial automorphisms of the complex of curves is the mapping class group.
The proof that the natural map \(\text{Out}(F_n)\to\operatorname{Aut}(K_n)\) is an isomorphism centers on viewing the complex \(K_n\) as the geometric realization of a poset of finite marked graphs and the link of each vertex of \(K_n\) as a wedge of \((2n-4)\)-dimensional spheres. A minimal vertex called a rose, \(\rho_0\), may be determined by calculating the number of spheres in the wedge. These calculations are used to show that any simplicial automorphism \(f\) of \(K_n\) must preserve the poset structure of \(K_n\).
By considering the star, \(st(\rho_0)\), of \(\rho_0\) and using techniques of Nielsen graphs, it is shown that if an automorphism \(f\) of \(K_n\) leaves \(st(\rho_0)\) invariant, then \(f\) acts on \(st(\rho_0)\) in the same manner as some element of the stabilizer of \(\rho_0\) in \(\text{Out}(F_n)\). Using the fact that any two roses in \(K_n\) can be connected by a sequence of roses that are Nielsen adjacent, it is proved that \(K_n\) is the union of the stars of roses. Several figures are used to clarify group theoretic concepts and to illustrate the transformations between the various Nielsen graphs in the link of a rose.

MSC:

20E05 Free nonabelian groups
20F65 Geometric group theory
20F28 Automorphism groups of groups
57M07 Topological methods in group theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. R. Bridson, “Geodesics and curvature in metric simplicial complexes” in Group Theory from a Geometrical Viewpoint , World Sci. Publishing, River Edge, N.J., 1991, 373–.\hs463. · Zbl 0844.53034
[2] M. R. Bridson and B. Farb, A remark about actions of lattices on free groups , to appear in J. Topol. Appl. · Zbl 0964.22011 · doi:10.1016/S0166-8641(99)00174-1
[3] M. R. Bridson and K. Vogtmann, Automorphisms of automorphisms of free groups , J. Algebra 229 (2000), 785–792. · Zbl 0959.20027 · doi:10.1006/jabr.2000.8327
[4] M. Culler, “Finite groups of outer automorphisms of free groups” in Contributions to Group Theory , Contemp. Math. 33 , Amer. Math. Soc., Providence, R.I., 1984, 197–207. · Zbl 0552.20024
[5] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups , Invent. Math. 84 (1986), 91–119. · Zbl 0589.20022 · doi:10.1007/BF01388734
[6] J. Dyer and E. Formanek, The automorphism group of a free group is complete , J. London Math. Soc. (2) 11 (1975), 181–190. · Zbl 0313.20021 · doi:10.1112/jlms/s2-11.2.181
[7] C. Earle and I. Kra, On isometries between Teichmüller spaces , Duke Math. J., 41 (1974), 583–591. · Zbl 0293.32020 · doi:10.1215/S0012-7094-74-04163-5
[8] B. Farb and H. Masur, Superrigidity and mapping class groups , Topology 37 (1998), 1169–1176. · Zbl 0946.57018 · doi:10.1016/S0040-9383(97)00099-2
[9] W. J. Harvey, “Geometric structure of surface mapping class groups” in Homological Methods in Group Theory (Durham, 1977) , London Math. Soc. Lecture Note Ser. 36 , Cambridge Univ. Press, Cambridge, 1979, 255–269. · Zbl 0424.57006
[10] N. V. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces , Internat. Math. Res. Notices 1997 , 651–.\hs666. · Zbl 0890.57018 · doi:10.1155/S1073792897000433
[11] K. Korkmaz, Complexes of curves and mapping class groups , Ph.D. dissertation, Michigan State Univ., 1996.
[12] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces , Ann. of Math. Stud. 78 , Princeton Univ. Press, Princeton, 1973. · Zbl 0265.53039
[13] H. L. Royden, “Automorphisms and isometries of Teichmüller spaces” in Advances in the Theory of Riemann Surfaces (Stony Brook, N.Y., 1969) , Ann. Math. Stud. 66 , Princeton Univ. Press, Princeton, 1971, 369–383. · Zbl 0218.32011
[14] J. Smillie and K. Vogtmann, “A generating function for the Euler characteristic of \(\Out(F_ n)\)” in Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Ill., 1985) , J. Pure Appl. Algebra 44 (1987), 329–348. · Zbl 0616.20009 · doi:10.1016/0022-4049(87)90036-3
[15] J. Tits, Buildings of Spherical Type and Finite \(BN\)-Pairs , Lecture Notes in Math. 386 , Springer, Berlin, 1974. · Zbl 0295.20047
[16] K. Vogtmann, Local structure of some \(\Out(F_ n)\)-complexes , Proc. Edinburgh Math. Soc. (2) 33 (1990), 367–379. · Zbl 0694.20021 · doi:10.1017/S0013091500004818
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.