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.


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


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