Światosław R. Gal

Euler characteristic of the configuration space of a complex

A closed form formula (generating function) for the Euler characteristic of the configuration space of

On the algebraic independence of Hamiltonian characteristic classes

\scriptstyle n

A two-cocycle on the group of symplectic diffeomorphisms

particles in a simplicial complex is given.

Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability

We prove that Hamiltonian characteristic classes defined as fibre integrals of powers of the coupling class are algebraically independent for generic coadjoint orbits.

Uniform symplicity of groups with proximal action

We investigate the properties of a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifold defined by Ismagilov, Losik, and Michor. We provide both vanishing and nonvanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.

Curvature spectra of simple Lie groups

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz–Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability are presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

A cocycle on the group of symplectic diffeomorphisms

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.

On approximation properties of semidirect products of groups

The Killing form β of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let Ω denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that Ωβ=2β. We observe that the result of Meyberg (in Abh. Math. Semin. Univ. Hamb. 54:177–189, 1984), describing the spectrum of Ω in complex simple Lie groups, easily leads to an analogous description for real simple Lie groups. In particular, 1 is not an eigenvalue of Ω in any real or complex simple Lie group G except those locally isomorphic to SL(

Finite index subgroups in Chevalley groups are bounded: an addendum to "On bi-invariant word metrics".

n,\mathbb {C}

\int_x^{hx}(g^*\alpha-\alpha)

) or one of its real forms. As shown in our recent paper (Derdzinski and Gal in Indiana Univ. Math. J., to appear), the last conclusion implies that, on such simple Lie groups G, nonzero multiples of the Killing form β are isolated among left-invariant Einstein metrics. Meyberg’s theorem also allows us to understand the kernel of Λ, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.