# 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}
}
• 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…
527 Citations
COMPARING LENGTH FUNCTIONS ON FREE GROUPS
The aim of this note is to study a pair of length functions on a free group, associated to a point in the unprojectivized Outer space. We discuss various characteristics which compare them and
Algebraic laminations for free products and arational trees
• Mathematics
Algebraic & Geometric Topology
• 2019
This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend
Cohomological dimension and symmetric automorphisms of a free group
Among a number of recent results concerning the cohomology of groups one of the most interesting is that obtained by Gersten [10] and Culler and Vogtmann [7] to the effect that if F is a free group
Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups
Stallings remarked that an outer automorphism of a free group may be thought of as a subdivision of a graph followed by a sequence of folds. In this thesis, we prove that automorphisms of fundamental
RATIONAL HOMOLOGY OF AUT ( F n )
• Mathematics
• 1998
In [7] a space An was introduced on which the group Aut(Fn) of automorphisms of a free group of rank n acts with finite stabilizers. An is a basepointed version of the “Outer space” introduced in [3]
Homological stability for automorphism groups of free groups
Let F, be a free group on n generators, Aut(F,) its group of automorphisms, and Out(F,) its outer automorphism group, the quotient of Aut(F,) by inner automorphisms. There has been much progress of
Automorphism groups of free groups, surface groups and free abelian groups
• Mathematics
• 2005
The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy
The boundary of the outer space of a free product
Let G be a countable group that splits as a free product of groups of the form G = G1 *···* Gk * FN, where FN is a finitely generated free group. We identify the closure of the outer space PO(G,
Morita classes in the homology of automorphism groups of free groups
• Mathematics
• 2004
Using Kontsevich's identification of the homology of the Lie algebra l1 with the co- homology of Out(Fr), Morita defined a sequence of 4k-dimensional classesk in the unstable rational homology of
On automorphisms of free groups and free products and their fixed points
Free group outer automorphisms were shown by Bestvina and Randell to have fixed subgroups whose rank is bounded in terms of the rank of the underlying group. We consider the case where this upper

## References

SHOWING 1-10 OF 11 REFERENCES
Equivalence of Elements Under Automorphisms of a Free Group
• Mathematics
• 1974
J. H. C. Whitehead [4,5] proved by topological means a theorem that enables one to decide whether the elements of a free group represented by two given words are equivalent under an automorphism of
On Equivalent Sets of Elements in a Free Group
together with the 'simple automorphism' which replaces a, by its inverse. Relative to this kind of equivalence we have very little to add to a paper by J. Nielsen,2 in which he gives a mechanical
Coinitial Grapfis and Whitehead Automorphisms
• A. Hoare
• Mathematics
Canadian Journal of Mathematics
• 1979
Coinitial graphs were used in [2; 3 ; 4] as a combinatorial tool in the Reidemeister- Schreier process in order to prove subgroup theorems for Fuchsian groups. Whitehead had previously introduced
Confluent and Related Mappings Defined by Means of Quasi-Components
In 1964, J. J. Charatonik in [1] introduced a new class of mappings, the so-called confluent mappings, which comprises the classes of open, monotone and quasi-interior mappings (see [20]). In 1966,
Arbres , Amalgames , SL 2
• 1977