Karen Vogtmann

Moduli of graphs and automorphisms of free groups

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 :

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.

CERF THEORY FOR GRAPHS

We develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn).

Homology stability for outer automorphism groups of free groups

We prove that the quotient map from Aut(Fn) to Out(Fn) induces an isomorphism on homology in dimension i for n at least 2i + 4. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for Aut(Fn), showing that its ith homology is independent of n for n at least 2i + 2. AMS Classification 20F65; 20F28, 57M07

A group theoretic criterion for property FA

We give group-theoretic conditions on a set of generators of a group G which imply that G admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups, including automorphism groups of most free groups and mapping class groups of most surfaces.

Automorphisms of 2-dimensional right-angled Artin groups

We study the outer automorphism group of a right-angled Artin group AA in the case where the defining graph A is connected and triangle-free. We give an algebraic description of Out.AA/ in terms of maximal join subgraphs in A and prove that the Tits’ alternative holds for Out.AA/. We construct an analogue of outer space for Out.AA/ and prove that it is finite dimensional, contractible, and has a proper action of Out.AA/. We show that Out.AA/ has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound. 20F36; 20F65, 20F28

Local structure of some Out ( F n )-complexes

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn)

We study a family of Lie algebras {hO} which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map which generalizes the trace map defined by S. Morita. For the Lie operad we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL(2,Z). Using cusp forms we construct new cycles for the unstable homology of Out(F_n), and using Eisenstein series we find new cycles for Aut(F_n). For the associative operad we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.

Homomorphisms from automorphism groups of free groups

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If m < n then a homomorphism Aut(Fn) → Aut(Fm) can have image of cardinality at most 2. More generally, this is true of homomorphisms from Aut(Fn) to any group G that does not contain an isomorphic image of the symmetric group Sn+1. Strong restricitons are also obtained on maps to groups which do not contain a copy of Wn = (Z/2) n o Sn, or of Z n−1. These results place constraints on how Aut(Fn) can act. For example, if n ≥ 3 then Aut(F3) has no non-trivial action on the cirlce (by homeomorphisms).

Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z_2-homology sphere of dimension less than n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces either. When n is even, we obtain similar results with Z_3 coefficients.