172 research outputs found
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
On subgroups of minimal topological groups
A topological group is minimal if it does not admit a strictly coarser
Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a
topological group is the greatest lower bound of the left and right
uniformities. A group is Roelcke-precompact if it is precompact with respect to
the Roelcke uniformity. Many naturally arising non-Abelian topological groups
are Roelcke-precompact and hence have a natural compactification. We use such
compactifications to prove that some groups of isometries are minimal. In
particular, if U_1 is the Urysohn universal metric space of diameter 1, the
group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact,
topologically simple and minimal. We also show that every topological group is
a subgroup of a minimal topologically simple Roelcke-precompact group of the
form Iso(M), where M is an appropriate non-separable version of the Urysohn
space.Comment: To appear in Topology and its Applications. 39 page
On a stronger reconstruction notion for monoids and clones
Motivated by reconstruction results by Rubin, we introduce a new
reconstruction notion for permutation groups, transformation monoids and
clones, called automatic action compatibility, which entails automatic
homeomorphicity. We further give a characterization of automatic
homeomorphicity for transformation monoids on arbitrary carriers with a dense
group of invertibles having automatic homeomorphicity. We then show how to lift
automatic action compatibility from groups to monoids and from monoids to
clones under fairly weak assumptions. We finally employ these theorems to get
automatic action compatibility results for monoids and clones over several
well-known countable structures, including the strictly ordered rationals, the
directed and undirected version of the random graph, the random tournament and
bipartite graph, the generic strictly ordered set, and the directed and
undirected versions of the universal homogeneous Henson graphs.Comment: 29 pp; Changes v1-->v2::typos corr.|L3.5+pf extended|Rem3.7 added|C.
Pech found out that arg of L5.3-v1 solved Probl2-v1|L5.3, C5.4, Probl2 of v1
removed|C5.2, R5.4 new, contain parts of pf of L5.3-v1|L5.2-v1 is now
L5.3,merged with concl of C5.4-v1,L5.3-v2 extends C5.4-v1|abstract, intro
updated|ref[24] added|part of L5.3-v1 is L2.1(e)-v2, another part merged with
pf of L5.2-v1 => L5.3-v
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