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We establish the following non-abelian analogue of the Fundamental Theorem of Projective Geometry: the natural map from ${\rm{Aut}}(F_n)$ to the automorphism group of the free-factor complex $\mathcal{AF}_n$ is an isomorphism. We also prove the corresponding theorem for the action of ${\rm{Out}}(F_n)$ on the complex of conjugacy classes of free factors.

We prove that if $G$ fibres algebraically and is part of a $\mathrm{PD}^3$-pair, then $G$ is the fundamental group of a fibred compact aspherical 3-manifold. This yields a new, homological proof of a classical theorem of Stallings: if $G = \pi_1(M^3)$ is the fundamental group of a compact irreducible 3-manifold $M^3$ and $\phi \colon G \to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $\phi$ is induced by a topological fibration of $M^3$ over the circle., Comment: 7 pages

For $N \geq 3$, the abstract commensurators of both ${{\rm{Aut}}}(F_N)$ and its Torelli subgroup ${{\rm{IA}}}_N$ are isomorphic to ${{\rm{Aut}}}(F_N)$ itself., Comment: 29 pages, 5 figures

The group $\mathfrak{X}(G)$ is obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. We make significant additions to the list of properties that the functor $\mathfrak{X}$ is known to preserve. We also investigate the geometry and complexity of the word problem for $\mathfrak{X}(G)$. Subtle features of $\mathfrak{X}$ are encoded in a normal abelian subgroup $W, Comment: 25 pages, no figures

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated., Comment: 12 pages, 2 figures. Final version to appear in Bulletin of the London Math Soc

We construct classes of K��hler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve., 21 pages, 1 figure, V2: Minor corrections and improvements to the exposition. Final accepted version, to appear in the Transactions of the American Mathematical Society

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of type ${\rm{FP}}_n$ if and only if it is of type ${\rm{F}}_n$. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri type. The template for these examples leads to a more constructive characterization of finitely presented residually free groups up to commensurability. We show that the class of finitely presented residually free groups is recursively enumerable and present a reduction of the isomorphism problem. A new algorithm is described which, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. The (multiple) conjugacy and membership problems for finitely presented subgroups of residually free groups are solved., Comment: 38 pages

We study the automorphisms \phi of a finitely generated free group F. Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative f:G\to G of a power of \phi that behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the Beaded Decomposition Theorem which describes the structure of paths in G obtained by repeatedly passing to f-images of an edge and taking subpaths. This decomposition is the key to adapting our proof of the quadratic isoperimetric inequality for $F\rtimes_\phi\mathbb Z$, with \phi positive, to the general case. To illustrate the wider utility of our topological normal form, we provide a short proof that for every w in F, the function $n\mapsto |\phi^n(w)|$ grows either polynomially or exponentially., Comment: Version 2 is 35 pages. The section on growth contained an error pointed out by Gilbert Levitt, and has been removed. A corrected verion will appear in a subsequent paper. Otherwise, changes are mostly cosmetic

If $F$ is a finitely generated free group and $\phi$ is a positive automorphism of $F$ then $F\rtimes_\phi Z$ satisfies a quadratic isoperimetric inequality., Comment: 69 pages, 32 figures. Improved proof of the Pincer Lemma