Аннотация:A topological group is a group with a topology with respect to which both group operations
(multiplication and inversion) are continuous. We shall consider only nondiscrete Hausdorff
group topologies, because only they are of interest. The existence of such a topology with certain
properties imposes constraints on the algebraic structure. Thus, there are groups which do not
admit any group topology at all, a compact group topology can exist only on residually finite
groups, and so on. One of the topological properties in worst agreement with the group structure
is extremal disconnectedness (a topological space is extremally disconnected if the closure
of any open set is open, or, equivalently, if its Boolean algebra of clopen subsets is complete),
and the class of groups admitting group topologies with the most diverse properties is that of
Boolean groups. The question of the existence in ZFC of an extremally disconnected topological
group, which was asked by Arhangel'skii in 1967, is still open, but it is known that any extremally
disconnected group contains an open Boolean subgroup (Malykhin, 1975). This question has proved
closely related to the existence of special ultrafilters on $\omega$, on the one hand, and
of nonclosed discrete sets in topological groups, on the other hand.
Boolean groups have the simplest structure; in particular, each Boolean group is a vector space
over the field $\mathbb F_2=\{0,1\}$ and is freely generated by any basis of this space. Thus, any
Boolean group is the free Boolean group $B(X)$ on a basis $X$ (that is, the set $[X]^{<\omega}$ of
all finite subsets of $X$ under the operation $\triangle$ of symmetric difference), and any
nondiscrete topology on $X$ induces the nondiscrete free Boolean group topology on $B(X)$ (the
weakest group topology such that any continuous map of $X$ to any Boolean topological group extends
to a continuous homomorphism). However, this is of little help in solving the extremal
disconnectedness problem:
\smallskip
\noindent
\emph{If there exists a nondiscrete topological space $X$ for which $B(X)$ is
extremally disconnected, then there exists a Ramsey ultrafilter on $\omega$.} (The nonexistence of
Ramsey ultrafilters is consistent with ZFC.)
\smallskip
One of the most natural topologies on a set $X$ is that generated by a filter: we take any point
$*\in X$ and any free filter $\mathcal F$ on $Y=X\setminus \{*\}$; the neighborhoods of $*$ are the
elements of $\mathcal F$, and all points of $Y$ are isolated. We denote the topological space thus
obtained by $X_{\mathcal F}$. The free Boolean topological group $B(X_{\mathcal F})$ is
topologically isomorphic to the quotient $B(X_{\mathcal F})/\{0, *\}$; we denote this quotient by
$B(\mathcal F)$.
\smallskip\noindent
\emph{An ultrafilter $\mathcal U$ on a countable Boolean group
is Ramsey iff it contains a linearly independent set $X$ and
$B({\mathcal U\restriction X})$ is extremally disconnected.}
\smallskip
\noindent
\emph{If an extremally disconnected countable Boolean group contains a nondiscrete linearly
independent set, then there exists a $P$-point ultrafilter on $\omega$.} (The nonexistence of
$P$-point ultrafilters is consistent with ZFC.)
\smallskip
\noindent
\emph{If there exists a Ramsey ultrafilter on a cardinal $\kappa$, then there exists a
(nondiscrete) Boolean topological group of cardinality $\kappa$ in which
all bases (and hence all independent sets) are closed and discrete.}
\smallskip
On the other hand, the existence of only one closed discrete basis in a countable Boolean
topological group does not require additional set-theoretic assumptions:
\emph{Any countable Boolean topological group has a closed discrete basis.}
Zelenyuk proved that the existence of any (not necessarily independent) nonclosed
discrete countable set in an extremally disconnected group implies that of $P$-point ultrafilters.
On the other hand, the nonexistence of (nondiscrete) countable topological groups without such sets
is consistent with ZFC:
\smallskip
\noindent
{\it If there exist no rapid (ultra)filters on $\omega$, then
\noindent\textup{(i)} \rightskip\parindent \vtop{\noindent any countable nondiscrete topological group contains
a discrete set with precisely one
limit point;}
\noindent\textup{(ii)} \rightskip\parindent \vtop{\noindent any countable nondiscrete Boolean topological group contains two
disjoint discrete subsets for each of which zero is the only limit point.}
}
\smallskip
It is unknown whether there exists a model of ZFC in which there are neither $P$-point
nor rapid ultrafilters; however, an extremally disconnected group cannot contain two
disjoint sets specified in (iixc). Therefore, the nonexistence of countable extremally disconnected
groups is consistent with ZFC.
The key role in the proof of the last theorem is played by a new class of large sets in groups
introduced by Reznichenko and the author and called vast sets. Various notions of
large sets in groups and semigroups naturally arise in dynamics and combinatorial number theory.
Most familiar are those of syndetic, thick (or replete), and piecewise syndetic sets. (A
set in a group is syndetic if finitely many translates of this set cover the group, a set is thick
if it intersects all syndetic sets, and piecewise syndetic sets are the intersections of syndetic
sets with thick ones.) It is hard to say which is more interesting, these sets themselves or the
interplay between them. Thus, piecewise syndetic sets in $\mathbb N$ are partition regular, contain
arbitrarily long arithmetic progressions, and admit an ultrafilter characterization; the difference
set of a syndetic set in a countable Abelian group almost (up to a set of upper Banach density
zero) contains a set open in the Bohr topology, and so on. Vast sets are unique in that they form a
filter (although they can generate a group topology only under additional set-theoretic
assumptions); on the other hand, vast sets in Boolean groups are very close to $\Delta^*_n$-sets
introduced by Bergelson, Furstenberg, and Weiss for $\mathbb Z$. We construct new examples
distinguishing between various kinds of large sets and characterize certain (in particular, arrow
and Ramsey) ultrafilters on arbitrary infinite sets in terms of vast sets in Boolean groups.