Automorphisms of the graph of free splittings

By Javier Aramayona, Juan Souto. 2009

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We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represented by free splittings admitting a common refinement. The group Out(Fn) of outer automorphisms of Fn acts simplicially on Gn. Denoting by Aut(Gn) the group of simplicial automorphisms of the free splitting graph, we prove: Theorem 1. The natural map Out(Fn)→ Aut(Gn) is an isomorphism for n ≥ 3. We briefly sketch the proof of Theorem 1. We identify Gn with the 1skeleton of the sphere complex Sn and observe that every automorphism of Gn extends uniquely to an automorphism of Sn. It is due to Hatcher [4] that the sphere complex contains an embedded copy of the spine Kn of Culler-Vogtmann space. We prove that the latter is invariant under Aut(Sn), and that the restriction homomorphism Aut(Sn) → Aut(Kn) is injective. The claim of Theorem 1 then follows from a result of Bridson-Vogtmann [1] which asserts that Out(Fn) is the full automorphism group of Kn. Before concluding this introduction we would like to point out that recently Martino and Francaviglia [9] have proved that Out(Fn) is also the full isometry group of Culler-Vogtmann space when the latter is endowed with the Lipschitz metric. While one could claim that Theorem 1 is the analogue of Ivanov’s theorem on the isometries of the curve complex [6], the result of Martino and Francaviglia is the analogue of Royden’s theorem on the isometries of Teichmuller space [10]. The first author was partially supported by M.E.C. grant MTM2006/14688 and NUI Galway’s Millennium Research Fund; the second author was partially supported by NSF grant DMS-0706878 and Alfred P. Sloan Foundation.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f55a7544ae65c7fa113d793ccbf6976e17a50470

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Citations in the corpus

The free splitting complex of a free group I: Hyperbolicity (2011, background): “Fn/ and Hatcher’s sphere complex can be found in Aramayona and Souto [1], and that proof extends with little trouble to the entire complexes.”
The hyperbolicity of the sphere complex via surgery paths (2012, background): “In particular, Section 8.2 has been motivated by a question of Karen Vogtmann and comments by Juan Souto and Saul Schleimer.”
On hyperbolicity of free splitting and free factor complexes (2012, background): “Aramayona and Souto [1] showed that every automorphism of FSN is induced by some element of Out(FN ).”