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The abstract of the talk (see below) may be downloaded in one of the following formats:
\newcommand{\M}{{\cal G}}
\newcommand{\T}{{\mathbb T}}
\global\def\hull#1{\langle{#1}\rangle}
Let $G$ be a topological group. The game $\mathfrak{G}(G)$ has two players $P1$
and $P2$.
Player P1 chooses an element $x\in G$ and player P2 tries to guess what element
$x$
has been chosen by $P1$ by asking questions of the following form. For a
sequences $\underline{m}=\{m_n\}$ of integers the question
$Q_{\underline{m}}$ is: does $x^{m_n}$ converge to 1 in $G$ when $n\to \infty$
?
The aim of the player P2 is to accumulate as much information about $x$ as
possible
knowing the answers of all $\mathfrak c$ many
questions $Q_{\underline{m}}$, i.e., the set $S(x)=\{\{m_n\}\in \Z^\omega:\lim
x^{m_n}=1\}$. Since $S(x)=S(x^{-1})$,
$P2$ can guess at most the {\em doubleton} $\{x, x^{-1}\}$, in case $x$ is
non-torsion,
or the set $\mbox{gen}(x)$ of all
generators of the cyclic group $\hull{x}$. In such a case we say that
$P2$ wins (in other words, P2 wins if $S(x)=S(y)$ always implies
$\hull{x}=\hull{y}$ in
$G$). We prove that:
\begin{itemize}
\item[(a)] P2 wins the game $\mathfrak{G}(G)$ for a non-discrete locally
compact
group $G$ iff $G$ is isomorphic to the circle group;
\item[(b)] P2 wins the game $\mathfrak{G}(G)$ for a discrete group $G$ iff
$G$
is isomorphic to a subgroup of the dicrete group $\Q/\Z$.
\end{itemize}
We also discuss the groups $G$ where the game $\mathfrak{G}(G)$ is most optimal
for player P1,
namely those with minimum information for P2 (i.e., $S(x)=S(y)$ whenever $x,y$
are
non-trivial elements of $G$, e.g., the reals $\R$, the group of $p$-adic
numbers, etc.).
We give a complete characterization of these groups $G$ in the class of locally
compact abelian groups. The proofs are based on the structure theory of locally
compact groups and properties of topologically
torsion elements (cf. [1]).
\begin{thebibliography}{MM}
\bibitem{D1}
D.~Dikranjan, {\it Answer to a question of Armacost on topologically
torsion elements}, Second Honolulu Conference on Abelian
Groups and Modules, July 25 -- August 1, 2001, Hawaii, Preprint.
\end{thebibliography}