Discrete automorphism groups of convex cones of finite type

We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras and do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.Comment: 30 pages, to appear in Compositio Mat

The efficiency of plankton in the utilization of the sun radiation [Translation from: Briroda, 12, 29-35, 1948]

The efficiency of utilisation of the sun's radiation by natural communities has not been properly demonstrated with what so far has been obtained of reliable values, and it represents a great interest in many respects. A systematic study of the biotic balance of lakes was done in the course of a succession of summers starting in 1932, extensive material was obtained, which permitted to compute a value fear the utilisation of the sun's radiation by plankton in lakes, and to compare this with corresponding values for marine plankton and terrestrial vegetation

On the nature of the Virasoro algebra

The multiplication in the Virasoro algebra $$[e_p, e_q] = (p - q) e_{p+q} + \theta \left(p^3 - p\right) \delta_{p + q}, \qquad p, q \in {\mathbf Z},$$ $$[\theta, e_p] = 0,$$ comes from the commutator $[e_p, e_q] = e_p * e_q - e_q * e_p$ in a quasiassociative algebra with the multiplication \renewcommand{\theequation}{$*$} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p + \left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q}, \vspace{3mm}\\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication in a quasiassociative algebra ${\cal R}$ satisfies the property \renewcommand{\theequation}{$**$} \be a * (b * c) - (a * b) * c = b * (a * c) - (b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and sufficient for the Lie algebra {\it Lie}$({\cal R})$ to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassociative algebra ${\cal R}$ becomes associative. Formula $(*)$ above also has a differential-variational counterpart

Infinitely many hyperbolic Coxeter groups through dimension 19

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.Comment: This is the version published by Geometry & Topology on 11 July 2006 (V2: typesetting correction

Rolling of Coxeter polyhedra along mirrors

The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.Comment: 20pages, 15 figure

All flat manifolds are cusps of hyperbolic orbifolds

We show that all closed flat n-manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)-orbifold.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-13.abs.htm

Universal Realisators for Homology Classes

We study oriented closed manifolds M^n possessing the following Universal Realisation of Cycles (URC) Property: For each topological space X and each integral homology class z of it, there exist a finite-sheeted covering \hM^n of M^n and a continuous mapping f of \hM^n to X such that f takes the fundamental class [\hM^n] to kz for a non-zero integer k. We find wide class of examples of such manifolds M^n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we prove that for each 4-dimensional oriented closed manifold N^4, there exists a mapping of non-zero degree of a hyperbolic manifold M^4 to N^4. This was conjectured by Kotschick and Loeh.Comment: 20 pages, 1 figure; in version 2 minor corrections are made, 4 bibliography items and 1 figure are adde

Noncoherence of some lattices in Isom(Hn)

We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.Comment: This is the version published by Geometry & Topology Monographs on 29 April 2008. V3: typographical correction

Complete families of commuting functions for coisotropic Hamiltonian actions

Let G be an algebraic group over a field F of characteristic zero, with Lie algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a Poisson variety and each irreducible G-invariant subvariety X\subset g^* carries the induced Poisson structure. We prove that there is a family of algebraically independent polynomial functions {f_1,...f_l} on X, which pairwise commute with respect to the Poisson bracket and such that l=(dim X+tr.deg F(X)^G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varieties of corank zero and 2.Comment: Changed presentatio