We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.  2001 Elsevier Science

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In [12] and [13], M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads "commutative," "associative" and "Lie." We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F r ) outlined by Kontsevich. In [4] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [3] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.

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

For a right-angled Artin group A Γ , the untwisted outer automorphism group U(A Γ ) is the subgroup of Out(A Γ ) generated by all of the Laurence-Servatius generators except twists (where a twist is an automorphisms of the form v → vw with vw = wv). We define a space Σ Γ on which U(A Γ ) acts properly and prove that Σ Γ is contractible, providing a geometric model for U(A Γ ) and its subgroups. We also propose a geometric model for all of Out(A Γ ) defined by allowing more general markings and metrics on points of Σ Γ .

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We prove that the quotient map from Aut(F n ) to Out(F n ) 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(F n ), showing that its ith homology is independent of n for n at least 2i + 2.

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