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The abstract of the talk (see below) may be downloaded in one of the following formats:
The 1966 results of Comfort and Ross revealed
pseudocompact groups
as one of
those objects for which the confluence of
algebra and topology
yields a richer structure. Pseudocompact groups have since
drawn the attention of many general topologists,
who find among them a setting amenable to their techniques.
The topic of this talk belongs to this tradition and
will be centered on
the following two
extremal phenomena involving a pseudocompact group
$(G,\mathcal{T})$:
\begin{enumerate}
\item Can $(G,\mathcal{T})$ be extremal in the sense that
no finer
group topology
$\mathcal{ S}\supset \mathcal{T}$ on $G$ is pseudocompact?
\item Can $(G,\mathcal{T})$ be extremal in the
sense that it contains no proper dense subgroup
which is pseudocompact?
\end{enumerate}
Research around problems of this sort began
with a paper by Comfort and Soundararajan \cite{comfsound},
and
took its present form with
the monograph
\cite{comfrobe88} where, among other results, it was proved
that compact Abelian groups are not extremal in either of the senses
1 or 2 above.
Yet the question about the existence of (any kind of) extremal
pseudocompact groups remains unsolved,
a number of particular cases show
that extremality is rather a pathological property
and support the conjecture (with no evidence, though)
that no pseudocompact Abelian group
is extremal in either sense.
The style of the talk will be chiefly expository. We shall attempt to outline
the main techniques employed to find
finer pseudocompact topologies and dense pseudocompact subgroups,
to describe the deep ties existing between the two extremal
properties,
and to show
which Abelian pseudocompact groups
are the (few) remaining candidates for extremality.
Some unpublished results obtained jointly
with W.~W.~Comfort will be included.
\begin{thebibliography}{1}
\bibitem{comfrobe88}
W.~W. Comfort and Lewis~C. Robertson.
\newblock Extremal phenomena in certain classes of totally bounded groups.
\newblock {\em Dissertationes Math. (Rozprawy Mat.)}, 272, 1988.
\bibitem{comfsound}
W.~W. Comfort and T.~Soundararajan.
\newblock Pseudocompact group topologies and totally dense subgroups.
\newblock {\em Pacific J. Math.}, 100(1), 61--84, 1982.
\end{thebibliography}