Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
AUTOMORPHISMS OF THE GRAPH OF FREE JAVIER ARAMAYONA & JUAN SOUTO
 

Summary: AUTOMORPHISMS OF THE GRAPH OF FREE
SPLITTINGS
JAVIER ARAMAYONA & JUAN SOUTO
Abstract. We prove that every simplicial automorphism of the
free splitting graph of a free group Fn is induced by an outer au-
tomorphism 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 ver-
tices 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 1-
skeleton 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

  

Source: Aramayona, Javier - Department of Mathematics, National University of Ireland, Galway

 

Collections: Mathematics