The quadratic isoperimetric inequality for mapping tori of free group automorphisms II: The general case

If F is a finitely generated free group and \phi is an automorphism of F then the mapping torus of \phi admits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.Comment: 73 page

Actions of higher-rank lattices on free groups

If $G$ is a semisimple Lie group of real rank at least 2 and $\Gamma$ is an irreducible lattice in $G$, then every homomorphism from $\Gamma$ to the outer automorphism group of a finitely generated free group has finite image.Comment: 11 pages, no figures. Final version. To appear in Compositio Mat

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $\Gamma$, will construct explicit finite models for the skeleta of $K(\Gamma,1)$ and hence compute the integral homology and cohomology of $\Gamma$.Comment: 21 pages, 4 figure

Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 202

Commensurations of Aut(FN){{\rm{Aut}}}(F_N) and its Torelli subgroup

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 figure

On the geometry of the free factor graph for Aut(FN){\rm{Aut}}(F_N)

Let $\Phi$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $\Sigma$ with one boundary component. We show that if $b \in \pi_1(\Sigma)$ is the boundary word, $\phi \in {\rm{Aut}}(\pi_1(\Sigma))$ is a representative of $\Phi$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle \phi, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $\pi_1(\Sigma)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case.Comment: 12 pages, 1 figure. To appear in GG

Diacetyl in Australian dry red wines and its significance in wine quality

The diacetyl content of 466 Australian dry red table wines ranged from less than 0.1 ppm to 7.5 ppm with a mean of 2.4 ppm. Malo-lactic fermentation had occurred in 71 per cent of the wines, which had a mean diacetyl level of 2.8 ppm. In wines which had not undergone malo-lactic fermentation the mean diacetyl level 1.3 ppm.Taste threshold tests showed that a difference of as little as 1 ppm could be detected in a light dry red wine containing 0.3 ppm diacetyl. In a full flavoured darker wine of higher quality containing 3 ppm the minimum detectable addition was 1.3 ppm.It is considered that diacetyl in amounts up to 2 to 4 ppm, depending on the wine, improved quality by adding complexity to the flavour. Above these levels the aroma of diacetyl became identifiable as such and resulted in a reduction in quality. The diacetyl content of a range of red table wines stored at 15° C showed a mean decrease of 19 per cent in diacetyl content in 4 months, 22 per cent in 8 months, 26 per cent in 12 months and 28 per c ent in 18 months

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm