Ultrafilters and topologies on groups

The set of all vdtrafilters on a topological group $\overline{\tau}$ $(G,\tau)$ is considered which converge to the identity. The set $\overline{\tau}$ with the \breve{C}ech-Stone topology and the Glazer operation of multiplication of ultranlteres turns out to be a compact space and besides a semigroup with the operation of multiphication continuous in the second argument. The semigroup $\overline{\tau}$ is used as a tool for studying the topological group-$(G,\tau)$. Basing on some description of minimal right ideals of the semigroup $\overline{\tau}$, we prove the next
Theorem. {\it If a neighborhood $W$ of the identity of a topological group is subjected to partitioning into finitely many subsets $W=A_1\cup\dots\cup A_k$, then there are a natural $i$ and a finite set $K\subseteq G$ such that $A_i^{-1}A_iK$ is a neighborhood of the identity.}
It is preven that commutativity of the Semigroup $\overline{\tau}$ implies total desconnectedness of the group $(G,\tau)$. For every group topology $\tau$, we construct the finest topology among those totally bounded with respect to $\tau$ (a generalization of the constructions by Weil and Bohr). New cardinal invariants for groups are introduced (namely, ultrarank and index of noncompactness) as well as some methods for their calculating.