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Commutators of diffeomorphisms of a manifold with boundary

Tomasz Rybicki — 1998

Annales Polonici Mathematici

A well known theorem of Herman-Thurston states that the identity component of the group of diffeomorphisms of a boundaryless manifold is perfect and simple. We generalize this result to manifolds with boundary. Remarks on C r -diffeomorphisms are included.

A note on generalized flag structures

Tomasz Rybicki — 1998

Annales Polonici Mathematici

Generalized flag structures occur naturally in modern geometry. By extending Stefan's well-known statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included.

Lie algebras of vector fields and codimension one foliations.

Tomasz Rybicki — 1990

Publicacions Matemàtiques

The main result is a Pursell-Shanks type theorem for codimension one foliations. This theorem can be viewed as a partial solution of a hypothetical general version of the theorem of Pursell-Shanks. Several propositions and lemmas on foliations are contained in the proof.

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki — 2012

Annales Polonici Mathematici

It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation G . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing C r diffeomorphism group G is simple iff the foliation [ G , G ] defined by [G,G] admits no proper minimal sets....

On the homeomorphism groups of manifolds and their universal coverings

Agnieszka KowalikTomasz Rybicki — 2011

Open Mathematics

Let H c(M) stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold M. It is shown that H c(M) is perfect and simple under mild assumptions on M. Next, conjugation-invariant norms on Hc(M) are considered and the boundedness of Hc(M) and its subgroups is investigated. Finally, the structure of the universal covering group of Hc(M) is studied.

On the flux homomorphism for regular Poisson manifolds

Rybicki, Tomasz — 1998

Proceedings of the 17th Winter School "Geometry and Physics"

Author’s abstract: “We introduce the concept of the flux homomorphism for regular Poisson manifolds. First we establish a one-to-one correspondence between Poisson diffeomorphisms close to i d and closed foliated 1-forms close to 0. This allows to show that the group of Poisson automorphisms is locally contractible and to define the flux locally. Then, by means of the foliated cohomology, we extend this local homomorphism to a global one”.

On admissible groups of diffeomorphisms

Rybicki, Tomasz — 1997

Proceedings of the 16th Winter School "Geometry and Physics"

The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the followingTheorem. Let ( M i , α i ) , i = 1 , 2 , be a geometric structure such that its group of automorphisms G ( M i , α i ) satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3’, 4, 5, 6 and 7, and M i is compact, or axioms 1, 2,...

An infinite dimensional version of the third Lie theorem

Rybicki, Tomasz — 2002

Proceedings of the 21st Winter School "Geometry and Physics"

The concept of evolution operator is used to introduce a weak Lie subgroup of a regular Lie group, and to give a new version of the third Lie theorem. This enables the author to formulate and to study the problem of integrability of infinite-dimensional Lie algebras. Several interesting examples are presented.

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