This book presents the relationship between ultrafilters and topologies on groups. It shows how
ultrafilters are used in constructing topologies on groups with extremal properties and how
topologies on groups serve in deriving algebraic results about ultrafilters.The contents of the
book fall naturally into three parts. The first comprising Chapters 1 through 5 introduces to
topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not
required. Constructions of some important topological groups are given. In particular that of
an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that
every infinite group admits a nondiscrete zero-dimensional topology in which all translations
and the inversion are continuous.In the second part Chapters 6 through 9 the Stone-Cêch
compactification G of a discrete group G is studied. For this a special technique based on the
concepts of a local left group and a local homomorphism is developed. One proves that if G is a
countable torsion free group then G contains no nontrivial finite groups. Also the ideal
structure of G is investigated. In particular one shows that for every infinite Abelian group
G G contains 22 G minimal right ideals.In the third part using the semigroup G almost
maximal topological and left topological groups are constructed and their ultrafilter
semigroups are examined. Projectives in the category of finite semigroups are characterized.
Also one shows that every infinite Abelian group with finitely many elements of order 2 is
absolutely -resolvable and consequently can be partitioned into subsets such that every coset
modulo infinite subgroup meets each subset of the partition.The book concludes with a list of
open problems in the field. Some familiarity with set theory algebra and topology is
presupposed. But in general the book is almost self-contained. It is aimed at graduate
students and researchers working in topological algebra and adjacent areas.
