• DOI:10.1007/BF01388734
• Corpus ID: 122869546

Moduli of graphs and automorphisms of free groups

@article{Culler1986ModuliOG,
  title={Moduli of graphs and automorphisms of free groups},
  author={Marc Culler and Karen Vogtmann},
  journal={Inventiones mathematicae},
  year={1986},
  volume={84},
  pages={91-119}
}

• M. Culler, K. Vogtmann
• Published 1 February 1986
• Mathematics
• Inventiones mathematicae

This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study mapping classes of surfaces. Let F, denote the free group of rank n. We will study the g roup Out(F,) of outer au tomorphisms of F, by studying its act ion on a space X, which is analogous to the Teichmtiller space of hyperbol ic metrics on a surface; the points of X, are metric structures on graphs with fundamental group F,. We begin by making this not ion precise. By a graph we shall mean a connected 1-dimensional CW-complex. The 0cells will be called nodes and the l-cells edges. The valence of a node x is the number of oriented edges which terminate at x, i.e. the min imum number of components of an arbitrarily small deleted ne ighborhood of x. An N-graph is a graph endowed with a metric such that each edge is locally isometric to an interval in l l and such that the distance between two points is the length of the shortest edge-path joining them. An N-graph is said to be minimal if it is not homotopy equivalent to any proper subgraph. Th roughou t this paper we will consider only ~,-graphs which are minimal and have no nodes of valence 2. (A minimal N-g raph cannot have nodes of valence 1). Fix a (topological) graph R o with one node and n edges, and choose an identification F--~rl(Ro). If G is an N-graph, then a homotopy equivalence g : R o ~ G is called a marking on G. We define two markings g l :Ro,G1 and g 2 : R o ~ G 2 to be equivalent if there exists an isometry i: GI~G 2 making the following diagram commute up to (free) homotopy :