Abstract:
The first proven abyssal record of Stenosemus exaratus (G.O. Sars, 1878) is presented on the basis of an ROV study in the Irish Sea. For the first time in situ images of the species and data on the environmental parameters are provided.

Abstract:
We describe a novel biotope at 633 to 762 m depth on a vertical wall in the Whittard Canyon, an extensive canyon system reaching from the shelf to the deep sea on Ireland’s continental margin. We explored this wall with an ROV and compiled a photomosaic of the habitat. The assemblage contributing to the biotope was dominated by large limid bivalves, Acesta excavata (mean shell height 10.4 cm), and deep-sea oysters, Neopycnodonte zibrowii, at high densities, particularly at overhangs. Mean density of N. zibrowii increased with depth, with densities of the most closely packed areas of A. excavata also increasing with depth. Other taxa associated with the assemblage included the solitary coral Desmophyllum dianthus, cerianthid anemones, comatulid crinoids, the trochid gastropod Margarites sp., the portunid crab Bathynectes longispina and small fish of the family Bythitidae. The scleractinian coral Madrepora oculata, the pencil urchin Cidaris cidaris and a species of Epizoanthus were also common. Prominent but less abundant species included the flytrap anemone Actinoscyphia saginata, the carrier crab Paramola cuvieri, and the fishes Lepidion eques and Conger conger. Observations of the hydrography of the canyon system identified that the upper 500 m was dominated by Eastern North Atlantic Water, with Mediterranean Outflow Water beneath it. The permanent thermocline is found between 600 and 1000 m depth, i.e., in the depth range of the vertical wall and the dense assemblage of filter feeders. Beam attenuation indicated nepheloid layers present in the canyon system with the greatest amounts of suspended material at the ROV dive site between 500 and 750 m. A cross-canyon CTD transect indicated the presence of internal waves between these depths. We hypothesise that internal waves concentrate suspended sediment at high concentrations at the foot of the vertical wall, possibly explaining the large size and high density of filter-feeding molluscs.

Abstract:
One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps in the standard development, including the self-normalizing property of Borel subgroups.

Abstract:
We classify all the symmetric integer bilinear forms of signature (2,1) whose isometry groups are generated up to finite index by reflections. There are 8595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin's enumeration of the strongly square-free cases. Our technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg's algorithm and our "method of bijections". We also correct a minor error in Conway and Sloane's definition of their canonical 2-adic symbol.

Abstract:
We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact groups for n < 7. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.

Abstract:
Sometimes a hyperbolic Kac-Moody algebra admits an automorphic correction, meaning a generalized Kac-Moody algebra with the same real simple roots and whose denominator function has good automorphic properties; these for example allow one to work out the root multiplicities. Gritsenko and Nikulin have formalized this in their theory of Lorentzian Lie algebras and shown that the real simple roots must satisfy certain conditions in order for the algebra to admit an automorphic correction. We classify the hyperbolic root systems of rank 3 that satisfy their conditions and have only finite many simple roots, or equivalently a timelike Weyl vector. There are 994 of them, with as many as 24 simple roots. Patterns in the data suggest that some of the non-obvious cases may be the richest.

Abstract:
We give a simplified proof of J. A. Wolf's classification of groups that can act freely and isometrically on a round sphere of some dimension. We also remove a small amount of redundancy from his list. The classification is the same as the classification of Frobenius complements in finite group theory.

Abstract:
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid groups of various simple orbifolds. The cases D_n, tilde{B}_n, tilde{C}_n and tilde{D}_n are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except tilde{A}_n the quotient is finite. We illustrate the value of our braid calculus by performing with pictures a nontrivial calculation in the Artin groups of type D_n.

Abstract:
We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide new finite-covolume reflection groups acting on complex and quaternionic hyperbolic spaces. Specifically, we provide groups acting on CH^n for all n<6 and n=7, and on HH^n for n=1,2,3 and 5. We compare our groups to those discovered by Deligne and Mostow and show that our largest examples are new. For many of these Lorentzian lattices we show that the entire symmetry group is generated by reflections, and obtain a description of the group in terms of the combinatorics of a lower-dimensional positive-definite lattice. The techniques needed for our lower-dimensional examples are elementary, but to construct our best examples we also need certain facts about the Leech lattice. We give a new and geometric proof of the classifications of selfdual Eisenstein lattices of dimension < 7 and of selfdual Hurwitz lattices of dimension < 5.

Abstract:
We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the Y555 diagram. The group is the full isometry group of a certain lattice of signature (13,1) over the Eisenstein integers Z[cube root of 1]. Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over this ring.