We show that a non-trivial finitely generated residually finite rationally solvable (or RFRS) group G is virtually fibred, in the sense that it admits a virtual surjection to Z with a finitely generated kernel, if and only if the first L 2 -Betti number of G vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of 3-manifolds.

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincaré duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the L 2 -torsion polytope of Friedl-Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lück-Tillmann.

We investigate Friedl-Lück's universal L 2 -torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fnby-Z groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for 3-manifold groups.We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander seminorms defined by Harvey. The same inequalities are known to hold for 3manifold groups.We also prove that the Newton polytopes of the universal L 2 -torsion of a descending HNN extension of F 2 locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of F 2 has finitely many connected components.When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in GL(n, Z), we show that the Newton polytope of the universal L 2 -torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

Abstract. We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes, and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.

Abstract. We study homomorphisms between Out(F n ) and Out(F m ) for n 6 and m < n 2 , and conclude that if m = n then each such homomorphism factors through the finite group of order 2. In the course of the argument linear representations of Out(F n ) in dimension less than n+1 2 over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection Out(F n ) → GL n (Z) coming from the action of Out(F n ) on the abelianisation of F n . We obtain similar results about linear representation theory of Out(F 4 ) and Out(F 5 ).