ИСТИНА

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.