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Some universal Polish groups and group structures on the Urysohn space
- Speaker(s)
- Michal Doucha
- Affiliation
- IMPAN
- Date
- March 19, 2014, 4:15 p.m.
- Room
- room 5050
- Seminar
- Topology and Set Theory Seminar
I will present a construction of a
metrically universal abelian separable metric group, i.e., every other
separable abelian metric group embeds by an isometric monomorphism. This
answers the question and extends the results of Shkarin and Niemiec, who
constructed (independently) an abelian separable metric group which is only
topologically universal (for the class of abelian Polish
groups).
I will also discuss group
structures on the Urysohn universal space. Cameron and Vershik were the first
who found a structure of an abelian (monothetic) group on the Urysohn space.
Recently, Niemiec proved that the topologically universal group (constructed
independently by him and Shkarin) is also isometric to the Urysohn space. I
will show that the metrically universal abelian group is also a new example of
a group structure on the Urysohn space substantially different from the
Vershik-Shkarin-Niemiec examples. I will also present a non-abelian metric
group isometric to the Urysohn space with a conjecture that this group is
a universal (topologically or maybe even metrically) Polish group admitting
two-sided invariant metric.
There are also a lot of open
problems provided by the people involved (Vershik, Niemiec, Shkarin and
others).
