Tomasz Rybicki

On the Group of Diffeomorphisms Preserving a Locally Conformal Symplectic Structure

The automorphism group of a locally conformal symplectic structure is studied. It is shown that this group possesses essential features of the symplectomorphism group. By using a special type of cohomology the flux and Calabi homomorphisms are introduced. The main theorem states that the kernels of these homomorphisms are simple groups (for the precise statement, see Section 7). Some of the methods used may also be interesting in the symplectic case.

Reduction for locally conformal symplectic manifolds

Abstract It is shown how one can do symplectic reduction for locally conformal symplectic manifolds, especially with an action of a Lie group. This generalizes well-known procedures for symplectic manifolds to the slightly larger class of locally conformal symplectic manifolds. The whole setting is very conformally invariant.

Pseudo-n-Transitivity of the Automorphism Group of a Geometric Structure

We introduce a notion of pseudo-n- transitivity which is a nontransitive counterpart of the n-transitivity. The main result states that any group of diffeomorphisms which satisfies the locality condition is pseudo-n-transitive for each n ≥ 1.

Isomorphisms Between Groups of Homeomorphisms

We give simple conditions which ensure that the topological structure of a manifold is completely determined by a group of homeomorphisms. The reasonings are still valid in the smooth category.

Locally continuously perfect groups of homeomorphisms

The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case of equivariant homeomorphism group and other examples are also considered.

n-transitivity of bisection groups of a Lie groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ ℕ by an arbitrary group of Cr-bisections of a Lie groupoid Γ of class Cr, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.