# Papers and preprints

###### Click on the title of a paper to see an abstract:

## The boundary of bordified Outer space, arxiv:2404.11371

We study the boundary of the "Jewel space" J_n constructed in arXiv:1709.01296. This is an equivariant deformation retract of Outer space CV_n on which Out(F_n) acts properly and cocompactly, and is homeomorphic to the Bestvina-Feighn bordification of CV_n. In the current paper we analyze the structure of the boundary of J_n. We then use the desctiption of the simplicial closure CV_n^* as the sphere complex of a connected sum of n copies of S^1×S^2 to prove that this boundary is homotopy equivalent to the subcomplex of CV_n^* spanned by vertices at infinity.

## Finite subgroups of outer automorphism groups of RAAGs I: untwisted subgroups (with C. Bregman and R. Charney), to appear in Algebraic and Geometric Topology. arXiv:2308.09222

For any right-angled Artin group A_G, Charney-Stambaugh-Vogtmann showed that the subgroup U^0(A_G) of Out(A_G) generated by Whitehead automorphisms and inversions acts properly and cocompactly on a contractible space K_G. In the present paper we show that any finite subgroup of U^0(A_G) fixes a point of K_G. This generalizes the fact that any finite subgroup of Out(F_n) fixes a point of Outer Space, and implies that there are only finitely many conjugacy classes of finite subgroups in U^0(A_G) ).

## The Euler characteristic of the moduli space of graphs, (with M. Borinsky), Adv. Math. 432 (2023), Paper No. 109290.

The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e^{−1/4}(n/e)n/(nlogn)^2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.

## The geometry and topology of automorphism groups of free groups, Lecture notes from C.I.R.M. workshop April 2022 (from notes taken by G. Crowe and P. Helms), to appear in Panorama Synthese, Societe Mathematique de France. availble here.

These are lecture notes from the introductory workshop to the IHP semester ``Groups acting on fractals, hyperbolicity and self-similarity", held at C.I.R.M. in April 2022. They contain an introduction to the study of the group Out(F_n) of outer automorphisms of a free group via its action on Outer space.

## The complex of free factors of a free group, (with A. Hatcher), March 2022, arXiv:2203.15602.

We show that the geometric realization of the partially ordered set of proper free factors in a finitely generated free group of rank n is homotopy equivalent to a wedge of spheres of dimension n − 2.

*The original version of this paper, published in 1998 in the Oxford Quarterly, contained an error, and the purpose of the present version is to provide a correction. The main results of the paper are unchanged. The error was in Lemma 2.3 which was used in the proof of Theorem 2.5. Unfortunately, the statement of Lemma 2.3 was false; it is replaced here with a new lemma which suffices to prove Theorem 2.5 after a few small changes in the proof. Section 2 has been reorganized to accommodate the changes, and we have taken the occasion to make some further clarifications in this section*.## Computing Euler characteristics using quantum field theory

*,*(with M. Borinsky), to appear in Séminaires et Congr\`es, Society Math. Fr., arXiv:2202.08739This paper explains how to use quantum field theory techniques to find formal power series that encode the virtual Euler characteristics of Out(F_n) and related graph complexes. Finding such power series was a necessary step in the asymptotic analysis of χ(Out(F_n)) carried out in the authors' previous paper.

## Outer space for RAAGs

*,*(with C. Bregman and R. Charney), Duke Math. J. 172 (2023), no. 6, 1033–1108.For any right-angled Artin group A_G we construct a finite-dimensional space O_G on which the group Out(A_G) of outer automorphisms of A_G acts properly. We prove that O_G is contractible, so that the quotient is a rational classifying space for Out(A_G). The space O_G blends features of the symmetric space of lattices in R^n with those of Outer space for the free group F_n. Points in O_G are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with A_G

## The Euler characteristic of Out(F_n), (with M. Borinsky), Commentarii Math. Helv. 95 (2020), no. 4, 703-748.

We prove that the rational Euler characteristic of Out(F_n) is always negative and its asymptotic growth rate is \Gamma(n-3/2)/\sqrt{2\pi}l\og^2n. This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the Lambert W-function and the zeta function.

## Cube complexes and abelian subgroups of automorphism groups of RAAGs, (with B. Millard), Math. Proc. Cambridge Philos. Soc. 170 (2021), no 3, 523-547, DOI

We construct free abelian subgroups of the group U(A_G) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AG) was previously studied by Charney, Stambaugh and the second author, who constructed a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of the principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

## On the bordification of Outer space, (with K.-U. Bux and P. Smillie), J. London Math Soc. 98, Issue 1 (2018), 12–34.

We give a simple construction of an equivariant deformation retract of Outer space which is homeomorphic to the Bestvina-Feighn bordification. This results in a much easier proof that the bordification is (2n-5)-connected at infinity, and hence that Out(F_n) is a virtual duality group.

## The topology and geometry of automorphism groups of free groups, Proceedings of the 2016 European Congress of Mathematicians, EMS Publishing House, Zurich (2018), 181-202.

In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs. Further impetus for using graphs to study automorphism groups of free groups came from the introduction of a space of graphs, now known as Outer space, on which the group Out(F_n) acts nicely. The study of Outer space and its Out(F_n) action continues to give new information about the structure of Out(F_n), but has also found surprising connections to many other groups, spaces and seemingly unrelated topics, from phylogenetic trees to cyclic operads and modular forms. In this talk I will highlight various ways these ideas are currently evolving.

## Tethers and homology stability for surfaces, (with A. Hatcher), Algebr. Geom. Topol. 17 (2017), no. 3, 1871–1916.

Homological stability for sequences of groups is often proved by studying the spectral sequence associated to the action of a typical group in the sequence on a highly-connected simplicial complex whose stabilizers are related to previous groups in the sequence. In the case of mapping class groups of manifolds, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. In this paper we focus on the case of surfaces and show that by using more refined geometric objects consisting of certain configurations of curves with arcs that tether these curves to the boundary, the stabilizers can be greatly simplified and consequently also the spectral sequence argument. We give a careful exposition of this program and its basic tools, then illustrate the method using braid groups before treating mapping class groups of orientable surfaces in full detail.

## Contractibility of Outer space: reprise, Hyperbolic Geometry and Geometric Group Theory, Advanced Studies in Pure Mathematics 73 (2017), Math. Soc. Japan, 265–280.

This note contains a newly streamlined version of the original proof that Outer space is contractible.

## Outer space for untwisted automorphisms of right-angled Artin groups, (with R. Charney and Nathaniel Stambaugh), Geom. Topol. 21 (2017), no. 2, 1131–1178.

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

## Assembling homology classes in automorphism groups of free groups, (with J. Conant, A, Hatcher and M. Kassabov), Commentarii Math. Helv. 91 (2016), 751-806.

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out(F_n) and Aut(F_n) from classes for groups Γ_{k,s}, where the Γ_{k,s} generalize Out(F_k)=Γ_{k,0} and Aut(F_k)=Γ_{k,1}. The symmetric group \Sigma_s acts on H_*(Γ_{k,s}) by permuting leaves, and for trivial rational coefficients we compute the \Sigma_s-module structure on H_*(Γ_{k,s}) completely for k\geq 2. Assembling these classes then produces all the known nontrivial rational homology classes for Aut(F_n) and Out(F_n) with the possible exception of classes for n=7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut(F_n) and Out(F_n) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

## GL(n, Z), Out(F_n) and everything in between: automorphism groups of RAAGs, London Mathematical Society Lecture Note Series 422: Groups St Andrews 2013, 105-127, Cambridge University Press, 2015.

These are notes from a series of three lectures given at Groups St. Andrews in 2013.

## Higher hairy graph homology, (with J. Conant and M. Kassabov), Geometriae Dedicata 176 (2015), 345-374.

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads Comm, Assoc and Lie we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph homology of the trace map defined in [1], as a symplectic representation.

## On the geometry of Outer space, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 27–46.

These notes from two talks at the Current Events session of the annual AMS meeting, giving an exposition of work of Algom-Kfir, Bestvina, and others on the Lipschitz metric on Outer space.

## Actions of arithmetic groups on homology spheres and acyclic homology manifolds, (with M.R. Bridson, F. Grunewald and B. Zimmermann), Math. Z. 276 (2014), no. 1-2, 387--395.

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n \geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is Z/2.

## Hairy graphs and the unstable homology of M(g,s), Out(F_n) and Aut(F_n), (with J. Conant and M. Kass- abov), J Topology (2013) 6(1): 119-153.

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.

## The Dehn functions of Aut(F_n) and Out(F_n) (with M. R. Bridson), Annales de l'institut Fourier, 62 no. 5 (2012), p. 1811-1817.

For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In this note we give a shorter, more direct proof of this last reduction.

## Abelian covers of graphs and maps between outer automorphisms of free groups, (with M. R. Bridson), Math. Ann. 353 (2012), no. 4, 1069-1102.

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.

## Subgroups and quotients of automorphism groups of RAAGs, (with R. Charney), Low-dimensional and Symplectic Topology - Proceedings of Symposia in Pure and Applied Math., Michael Usher ed University of Georgia, Editor - AMS, 2011.

We study subgroups and quotients of outer automorphism groups of right-angled Artin groups (RAAGs). We prove that for all RAAGS, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.

## What is Outer space?, Notices of the AMS 55, No. 7 (2008) 784-786.

This is a 2-page introduction to Outer Space, in the What IS...? series of the Notices of the AMS.

## Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups, (with K.-U. Bux and R. Charney), Groups, Geometry and Dynamics 3 (4) (2009) 541-554.

We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled Artin group with defining graph a tree, the bounds agree.

## Actions of automorphism groups of free groups on spheres and acyclic manifolds, (with Martin Bridson), Commentarii Math. Helv. 86 (1) (2011)

For n at least 3, let SAut(Fn ) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on Rn induces non-trivial actions of SAut(Fn ) on Rn and on Sn-1. We prove that SAut(Fn ) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(Fn ) 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.

## Automorphisms of higher-dimensional right-angled Artin groups, (with Ruth Charney), Bull. London Math Soc. (41) February (2009) 94-102.

We study the algebraic structure of the automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved in [ChCrVo] for two-dimensional right-angled Artin groups.

## A presentation for Aut(F_n), (with H. Armstrong and B. Forrest), J. of Group Theory 11 (2008), 267- 276.

We study the action of the group Aut(F_n) of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method described by K. S. Brown to deduce a new presentation of Aut(F_n).

## Automorphism groups of right-angled Artin groups, (with Ruth Charney), Guido's book of conjectures, L'Enseignement Mathematiques Monographie No. 40, 2008.

This short note asks a couple of questions about automorphisms of right-angled Artin groups. Question 1.3 was answered in [ChV].

## Automorphisms of two-dimensional right-angled Artin groups, (with Ruth Charney and John Crisp), Geometry & Topology 11 (2007) 2227-2264.

We study the outer automorphism group of a right-angled Artin group A_G in the case where the defining graph G is connected and triangle-free. We give an algebraic description of Out(AG) in terms of maximal join subgraphs in G and prove that the Tits' alternative holds for Out(A_G). We construct an analogue of outer space for Out(A_G) and prove that it is finite dimensional, contractible, and has a proper action of Out(A_G). We show that Out(A_G) 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.

## Morita classes in the homology of Aut(F_n) vanish after one stabilization, (with Jim Conant), Groups, Geometry and Dynamics 2 (1) (2008) 121-138.

There is a series of cycles in the rational homology of the groups Out( Fn ), first discovered by S. Morita, which have an elementary description in terms of finite graphs. The first two of these give nontrivial homology classes, and it is conjectured that they are all nontrivial. These cycles have natural lifts to the homology of Aut( Fn ), which is stably trivial by a recent result of Galatius. We show that in fact a single application of the stabilization map Aut(Fn) \to Aut(Fn+1) kills the Morita classes, so that they disappear immediately after they appear.

## The cohomology of automorphism groups of free groups, Proceedings of the International Congress of Mathematicians, Madrid 2006, European Mathematical Society Publishing House, Zurich, 2006.

There are intriguing analogies between automorphism groups of finitely generated free groups and mapping class groups of surfaces on the one hand, and arithmetic groups such as GL(n, Z) on the other. We explore aspects of these analogies, focusing on cohomological properties. Each cohomological feature is studied with the aid of topological and geometric constructions closely related to the groups. These constructions often reveal unexpected connections with other areas of mathematics.

## Erratum to: Homology stability for outer automorphism groups of free groups,(with A. Hatcher and N.Wahl), Algebr. Geom. Topol. 6 (2006), 573–579.

Erratum

## Automorphisms of freegroups, surfacegroups and freeabelian groups, (withMartinBridson), in Problems on Mapping Class Groups and Related Topics, ed. by B. Farb, Proc. Symp. Pure Math. 74, Amer. Math. Soc., Providence RI, (2006).

In this article we describe some of the more striking features of the deep analogy between GL(n,Z), automorphism groups of free groups, and mapping class groups. Our main purpose is to present a list of open questions motivated by this analogy.

## Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Com- binatorial Group Theory, Part I (Haifa, 2000). Geom. Dedicata 94 (2002), 1–31.

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.

## Geodesics in the space of trees, informal notes

In a paper with Billera and Holmes published in 2001 we studied the space of finite metric trees with a fixed number of leaves, which can be interpreted as phylogenetic trees with positive branch lengths. That paper included some information about how to find the geodesic path between two trees and the corresponding distance, but did not include a specific algorithm for doing so. The current notes address this task. They grew out of conversations with John Smillie, and were written shortly after the Billera-Holmes-Vogtmann paper appeared. They were never published, partly because I believe that a little more work (which I haven't done) could produce a substantially better algorithm. Meanwhile several people have requested these notes, so I decided to make them generally available. here.

## Geometry of the space of phylogenetic trees (with L. Billera and S. Holmes), Advances in Applied Math 27 (2001), 733-767.

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.

## Automorphisms of automorphism groups of free groups (with M. R. Bridson), J. Algebra 229 (2000), no. 2, 785–792.

For n > 2, the groups Out(Fn ) and Aut(Fn ) are complete, i.e. they are centreless and all of their automorphisms are inner.

## The symmetries of Outer space (with M. R. Bridson), Duke Math Journal 106 (2001), 391–409.

For n>2, the natural map Out(Fn) \to Aut(Kn) from the outer automorphism group of the free group of rank n to the group of simplicial automorphisms of the spine of outer space is an isomorphism.

## A Whitehead algorithm for surface groups (with G. Levitt), Topology 39 (2000), no. 6, 1239–1251.

For G the fundamental group of a closed surface, we produce an algorithm which decides whether there is an element of the automorphism group of G which takes one specified finite set of elements to another. The algorithm finds such an automorphism if it exists. The methods are geometric and also apply to surfaces with boundary.

## An Equivariant Whitehead algorithm and conjugacy for roots of Dehn twists (with S. Krstic and M. Lustig), Proceedings of the Edinburgh Math Society 44 (2001), 117-141.

Given finite sets of cyclic words {u1,...,uk} and {v1,...,vk} in a finitely generated free group F and two finite groups A and B of outer automorphisms of F we produce an algorithm to decide whether there is an automorphism which conjugates A to B and takes ui to i for each i. If A and B are trivial, this is the classic algorithm due to J.H.C. Whitehead. We use this algorithm together with Cohen and Lustig's solution to the conjugacy problem for Dehn twist automorphisms of F to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of F which have linear growth.

## Rational homology of Aut(F_n) (with A. Hatcher), Mathematical Research Letters 5 (1998), 759-780.

We compute the rational homology in dimensions less than seven of the group of automorphisms of a finitely generated free group of arbitrary rank. The only non-zero group in this range is H_4(Aut(F_4 );Q), which is one-dimensional.

## The complex of free factors of a free group (with A. Hatcher), Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 196, 459–468.

We show that the geometric realization of the partially ordered set of proper free factors in a finitely generated free group of rank n is homotopy equivalent to a wedge of spheres of dimension n-2.

## Cerf theory for graphs (with A. Hatcher), J. London Math. Soc., 58, part 3 (1998), 633-655.

This paper develops a deformation theory for k-parameter families of pointed marked graphs with fixed fundamentalgroup F_n. Applications include a simple geometric proof of stability of the rational homology of Aut(F_n), computations of the rational homology of Aut(F_n) 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).

## A group-theoretic criterion for property FA (with M. Culler), Proc. Amer. Math. Soc. 124 (1996), no. 3, 677-683

We give group-theoretic conditions on a set of generators of a group G which imply that G admits no non-trivial actions 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.

## Homology stability for outer automorphism groups of free groups, (with Allen Hatcher), Algebr. Geom. Topol. 4 (2004), 1253-1272.

We prove that the quotient map from the automorphism group to the outer automorphism group of the free group of rank n is an isomorphism on homology in dimension i for n at least 2i+4. This corrects an earlier proof by the first author and 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 boundary. 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.

## Morita classes in the homology of automorphism groups of free groups, (with Jim Conant), Geom. Topol. 8 (2004), 1471-1499.

Using Kontsevich's identification of the homology of the Lie algebra l_\infty with the cohomology of Out(Fr ), Morita defined a sequence of 4k-dimensional classes µk in the unstable rational homology of Out(F2k+2 ). He showed by a computer calculation that the first of these is non-trivial, so coincides with the unique non-trivial rational homology class for Out(F_4 ). Using the 'forested graph complex' introduced in [2], we reinterpret and generalize Morita's cycles, obtaining unstable cycles for every odd-valent graph. The description of Morita's original cycles becomes quite simple in this interpretation, and we are able to show that the second Morita cycle also gives a nontrivial homology class. Finally, we view things from the point of view of a different chain complex, one which is associated to Bestvina and Feighn's bordification of outer space. We construct cycles which appear to be the same as the Morita cycles constructed in the first part of the paper. In this setting, a further generalization becomes apparent, giving cycles for objects more general than odd-valent graphs. Some of these cycles lie in the stable range. We also observe that these cycles lift to cycles for Aut(F_r ).

## Cut vertices in commutative graphs, (with James Conant and Ferenc Gerlits), Quart. J. Math 56(3) (2005), 321-336.

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This graph homology is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give two different proofs (one algebraic, one geometric) that the commutative graph complex is quasi-isomorphic to the quotient complex obtained by modding out by graphs with cut vertices. This quotient complex has the advantage of being smaller and hence more practical for computations. In addition, it supports a Lie bialgebra structure coming from a bracket and cobracket we defined in a previous paper. As an application, we compute the rational homology groups of the commutative graph complex up to rank 7.

## On a theorem of Kontsevich, (with Jim Conant), Algebr. Geom. Topol. 3 (2003) 1167-1224.

In two seminal papers 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 [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209 (2003), 219-230] 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.

## Infinitesimal operations on graph complexes, (with James Conant), Math. Ann. 327 (2003), 545–573.

In two seminal papers Kontsevich used a construction called graph homology as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich's graph complex construction was generalized to the context of operads by E. Getzler and M. Kapranov and by M. Markl. In [CoV], we show that Kontsevich's original results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to the entire connected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting constraints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F_n).

## Homomorphisms from automorphism groups of free groups, (with Martin R. Bridson), Bull. London Math. Soc. 35 (2003), no. 6, 785–792.

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 ) to 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 G which do not contain a copy of Wn (the semidirect product of (Z/2)n and Sn ), or of a free abelian group of rank n - 1. These results place constraints on how Aut(Fn ) can act. For example, if n > 2 then Aut(Fn ) has no non-trivial action on the cirlce (by homeomorphisms).

- <details">On the geometry of the group of automorphisms of a free group (with M. Bridson), Bull. London Math. Soc. 27 (1995), 544-552.
The groups Aut(F_3) and Out(F_3) satisfy strictly exponential isoperimetric inequalities; in particular they are not automatic. For n\geq 3, Aut(F_n) and Out(F_n) do not admit bounded bicombings of subexponential length, hence they cannot act properly and cocompactly by isometries on any simply-connected space of non-positive curvature, and they are not biautomatic.

## Isoperimetric inequalities for automorphism groups of free groups (with A. Hatcher), Pacific J. Math. 173 (1996), no. 2, 425-441.

In this paper we show that the groups of automorphisms and outer automorphisms of a finitely generated free group have isoperimetric functions which are bounded above by exponential functions. This exponential bound is best possible if the free group has rank three, but the best bound remains unknown in higher rank. Our techniques show more generally that n-dimensional isoperimetric functions for these groups are at most exponential for all n. A variation of the technique gives an asynchronous bounded combing of the mapping class group of a bounded surface.

## End invariants of the group of outer automorphisms of a free group, Topology 14 (1995) no. 3, 533-545.

We prove Out(F_n) is simply-connected at infinity for n at least 5.

## Length functions and outer space (with J. Smillie), Michigan Math J. 39 (1992) 485-493.

The embedding of Outer space into the space of projective length functions on F_n is not determined by any finite set of words.

## Equivariant Outer space and automorphisms of free-by-finite groups (with S. Krstic), Comment. Math. Helvetici 68 (1993) 216-262.

This paper constructs an outer space for automorphisms of free-by-finite groups and proves that it is contractible. This is equivalent to showing the fixed point set of any finite group of outer automorphisms of a free group acting on Outer space is contractible, i.e. Outer space is an E underbar G.

## Automorphisms of SL2 of imaginary quadratic integers, (with J. Smillie) Proc. A.M.S. 112 (1991) no. 3

We determine the outer automorphism groups of the two-dimensional special linear and projective special linear groups of the ring of integers in an imaginary quadratic field.

## The boundary of outer space in rank two, (with M. Culler) in Arboreal Group Theory (R. Alperin, ed), New York: Springer-Verlag (1991) 189-229.

Outer space embeds in the space of projective length functions on F_n, and its closure is compact. In this paper we give an explicit description of the closure in the case n=2. In particular, we show that the closure is contractible and give an imbedding of the closure as a two-dimensional subset of R^3 which makes it clear that the closure is an ANR.

## Local structure of some Out(F_n)-complexes, Proc. Edinburgh Math Soc. 33 (1990), 367-379.

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 act with finite stabilizers and finite quotient. In this paper, it is shown that these complexes re 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.

## Automorphisms of graphs, p-subgroups of Out(F_n) and the Euler characteristic of Out(F_n), (with J. Smil- lie) J. Pure and Appl. Algebra 49 (1987), 187-200.

Finite subgroups of Out(F_n) are studied by analyzing isometry groups of graphs. The results of this analysis are used to derive information about the powers of primes which devide the denominator of the rational Euler characteristic of Out(F_n). This information together with a result from a previous paper by the same authors gives information about the kernel of the map from Out(F_n) to GL(n,Z) for n even.

## A generating function for the Eule rcharacteristic of Out(F_n), (withJ.Smillie), J.Pure and Appl. Algebra 44 (1987), 329-348.

We give a generating function for the rational Euler characteristic of Out(F_n), using the action of Out(F_n) on the spine of Outer space.

## Moduli of graphs and automorphisms of free groups, (with M. Culler) Inventionnes 84 (1986), 91-119.

We introduce a space X_n of marked metric graphs on which the group Out(F_n) of outer automorphisms of a finitely-generated free group acts properly. We prove that X_n is contractible, and that it has an equivariant deformation retract K_n whose quotient is compact. We use K_n to find the virtual cohomological dimension of Out(F_n). The space X_n can be thought of as an analog of the Teichmuller space of marked hyperbolic surfaces with its action by the mapping class group of the surface.

## Rational homology of Bianchi groups, Math. Ann. 272 (1985), 399-419.

We use the reduction theory of Mendoza to calculate the rational homology of the Bianchi groups SL(2,O_{-d}) for small values of d, which gives a complete list of values of d for which the cuspidal cohomology vanishes.

## The integral homology of SL2 and P SL2 over Euclidean imaginary quadratic integers, (with J. Schwermer) Comment.Math.Helv. 58(1983)no.4,573-598.

We use a complex defined by Mendoza to completely calculate the cohomology of the Bianchi groups SL(2,O_{-d}) for the Euclidean cases d=1,2,3,7 and 11.

## A Stieffel complex for the orthogonal group of a field, Comment. Math. Helv. 57 (1982), no.1, 11-21.

In this paper we show that the poset of orthogonal frames in (F^n,n(1)) with at most k elements is (k-1)-spherical if n is sufficiently large. Here n(1) is the identity form, and F may be any field with finite pythagoras number, e.eg. a local or global field, finite field of real-closed field. We then use this poset to show that for n large with respect to m, the inclusion O_n \to O_m induces an isomorphism H_m(O_n) \to H_m(O_{n+1}), where homology is taken with integral coefficients.

## Spherical Posets and homology stability for On,n, Topology 20 (1981), 119-132.

Let F be a field with more than 2 elements, We prove that the map H_k(O_{n,n}(F))\to H_k(O_{n+1,n+1}(F)) induced by inclusion is an isomorphism for n\geq 3k

*Homology Stability for On,n*, Comm. Alg., 7 (1979), no.1., 9-38.For F a field of characteristic zero we show that the maps H_k(O_{n,n}(F))\to H_k(O_{n+1,n+1}(F)) and H_k(Sp(n,F))\to H_k(Sp(n+1,F)) induced by inclusion are surjective for n \geq 3k+1 and isomorphisms for n \geq 3k+3.