The symmetries of outer space.
type: Item
instance of: scholarly article
title: The symmetries of outer space.
zbMATH Open document ID: 1037.20025
DOI: 10.1215/S0012-7094-01-10625-X
author: Martin R. Bridson
author: Karen Vogtmann
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 1820781
zbMATH Keywords: outer automorphism groups
zbMATH Keywords: spines
zbMATH Keywords: Nielsen graphs
zbMATH Keywords: simplicial complexes
zbMATH Keywords: minimal vertex
zbMATH Keywords: stars of roses
zbMATH Keywords: free groups
zbMATH Keywords: outer space
zbMATH Keywords: mapping class groups
zbMATH Keywords: simplicial automorphisms
reviewed by: Raymond F. Tennant
MaRDI profile type: Publication
cites work: Q4865842
cites work: A remark about actions of lattices on free groups
cites work: Automorphisms of automorphism groups of free groups
cites work: Q3344126
cites work: Moduli of graphs and automorphisms of free groups
cites work: The Automorphism Group of a Free Group is Complete
cites work: On isometries between Teichmüller spaces
cites work: Superrigidity and mapping class groups
cites work: Q3858951
cites work: Q4365977
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Q5622542
cites work: A generating function for the Euler characteristic of \(Out(F_ n)\)
cites work: Buildings of spherical type and finite BN-pairs
cites work: Local structure of some Out(Fn)-complexes
full work available at URL: s0012-7094-01-10625-x
OpenAlex ID: W2023541794
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description: scientific article; zbMATH DE number 1820781