The symmetries of outer space.
The authors investigate the natural map \(\text{Out}(F_n)\to\Aut(K_n)\) 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 \(\Aut(K_n)\) 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 \(\text{Out}(F_n)\to\Aut(K_n)\) 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.
scientific article; zbMATH DE number 1820781
scientific article