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Ambitable topological groups
A topological group is said to be ambitable if each uniformly bounded
uniformly equicontinuous set of functions on the group with its right
uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n
bounded topological group is either precompact or ambitable. In the familiar
semigroups constructed over ambitable groups, topological centres have an
effective characterization.Comment: LaTeX; 12 pages; Changes in versions 2 and 3: New Theorem 3.3 and
improvements enabled by it; Changes in version 4: Corrections for Theorems
4.9 and 4.10, added 1 referenc
Compactifications of topological groups
Every topological group has some natural compactifications which can be a
useful tool of studying . We discuss the following constructions: (1) the
greatest ambit is the compactification corresponding to the algebra of
all right uniformly continuous bounded functions on ; (2) the Roelcke
compactification corresponds to the algebra of functions which are both
left and right uniformly continuous; (3) the weakly almost periodic
compactification is the envelopping compact semitopological semigroup of
(`semitopological' means that the multiplication is separately continuous).
The universal minimal compact -space is characterized by the
following properties: (1) has no proper closed -invariant subsets; (2)
for every compact -space there exists a -map . A group is
extremely amenable, or has the fixed point on compacta property, if is a
singleton. We discuss some results and questions by V. Pestov and E. Glasner on
extremely amenable groups. The Roelcke compactifications were used by M.
Megrelishvili to prove that can be a singleton. They can be used to
prove that certain groups are minimal. A topological group is minimal if it
does not admit a strictly coarser Hausdorff group topology.Comment: 17 page
Categorically closed topological groups
Let be a subcategory of the category of topologized semigroups
and their partial continuous homomorphisms. An object of the category
is called -closed if for each morphism
of the category the image is closed in . In the paper
we detect topological groups which are -closed for the categories
whose objects are Hausdorff topological (semi)groups and whose
morphisms are isomorphic topological embeddings, injective continuous
homomorphisms, continuous homomorphisms, or partial continuous homomorphisms
with closed domain.Comment: 26 page
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
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