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Ultrafilters and topologies on groups. (English) Zbl 1215.22001
de Gruyter Expositions in Mathematics 50. Berlin: de Gruyter (ISBN 978-3-11-020422-3/hbk; 978-3-11-021322-5/ebook). viii, 219 p. (2011).
For a discrete space $$X$$, we identify the Stone-Čech compactification $$\beta X$$ of $$X$$ with the set of all ultrafilters on $$X$$. If $$S$$ is a discrete semigroup then $$\beta S$$ has a rich algebraic structure and plenty of combinatorial applications, see [N. Hindman and D. Strauss, Algebra in the Stone-Čech compactification: Theory and Applications. De Gruyter Expositions in Mathematics. 27. Berlin: Walter de Gruyter (1998; Zbl 0918.22001)].
It is a commonplace that many topological notions can be expressed in terms of ultrafilters. If a semigroup $$S$$ is endowed with a topology $$\tau$$ which respects the algebra of $$S$$, some special subsemigroup of $$\beta S_{\text{discrete}}$$ could tell us much about $$(S,\tau)$$. On the other hand, an appropriate topologization of $$S$$ could be an effective tool in the study of $$\beta S_{\text{discrete}}$$. For the first topologizations of $$S$$ by means of $$\beta S$$ see [T. Papazyan, Topology Appl. 39, No. 3, 229–243 (1991; Zbl 0760.22003)], but the whole area was initiated in [I. V. Protasov, Sib. Math. J. 34, No. 5, 938–952 (1993); translation from Sib. Mat. Zh. 34, No. 5, 163–180 (1993; Zbl 0828.22002)]. Yevgen Zelenyuk is involved in the process since 1995, and this book is a scrupulous account based on his doctoral thesis defended at Kyiv University in 2000.
Chapter 1. Topological groups. For a countable group $$G$$ (a countable ring $$R$$) and a filter $$\mathcal{F}$$ on $$G$$ (on $$R$$) the author describes the strongest group (ring) topology in which $$\mathcal{F}$$ converges to the identity (zero) of $$G$$ (of $$R$$), and proves Markov’s criterion of the topologizability of a countable group and Arnautov’s theorem on the topologizability of each countable ring. The exposition follows [I. Protasov and Y. Zelenyuk, Topologies on groups determined by sequences. Mathematical Studies Monograph Series. 4. Lviv: VNTL Publishers. (1999; Zbl 0977.54029)]. Theorem 1.28 (on the strengthening of a group topology on a countable group to a group topology with countable base) is new.
Chapter 2. Ultrafilters. This is a standard background.
Chapter 3. Topological spaces with extremal properties. This is a translation of the standard definitions of maximal, submaximal, nodec, and irresolvable spaces in Ultro based on [Y. G. Zelenyuk and I. V. Protasov, ‘Dopovidi NAN Ukrain., 1997, No. 3, 7–11 (1997)]. For the case of a convergent sequence, the formally new Theorem 3.33 was proved by Malykhin.
Chapter 4. Left invariant topologies and strongly discrete filters. A filter $$\mathcal{F}$$ on an infinite group $$G$$ is called strongly discrete if $$\mathcal{F}$$ contains the Fréchet filter and there is a mapping $$M:G\to\mathcal{F}$$ such that the subsets $$\{xM(x): x\in G\}$$ are pairwise disjoint. It should be mentioned that a free ultrafilter on a countable group $$G$$ is strongly discrete if and only if it is a right cancellable in the semigroup $$\beta G$$. By Theorem 4.18, the strongest left invariant topology on $$G$$ in which $$\mathcal{F}$$ converges to the identity is Hausdorff. By Theorem 4.32, there is a strongly discrete filter $$\mathcal{F}$$ on $$G$$ such that $$\mathcal{F}=\mathcal{F}^{-1}$$ and $$x^{-1}\mathcal{F}x=\mathcal{F}$$ for each $$x\in G$$. As a corollary, we get a non-discrete Hausdorff topologization of $$G$$ in which each left shift, each right shift and the inversion $$x\to x^{-1}$$ are continuous. By Lemma 4.33, every finite system of inequalities of the form $$ayb\neq y^{\pm 1}$$, where $$ab\neq 1$$, has $$|G|$$-many solutions in $$G$$. In other words, the second Zariski topology $$\zeta_2(G)$$ on each infinite group $$G$$ is not discrete. On the other hand, a slight modification of the Adian group $$A(2,665)$$ gives $$G$$ with discrete $$\zeta_{665}(G)$$. What between 2 and 665, this is the question!
Chapter 5. Topological groups with extremal properties. The author modernizes the remarkable example of an extremally disconnected topological group [S. Sirota, Math. USSR, Sb. 8, 169–180 (1969); translation from Mat. Sb., n. Ser. 79(121), 179–192 (1969; Zbl 0193.51201)], replaces CH to $$\mathfrak{p}=\mathfrak{c}$$ in Malykhin’s construction of a maximal topological group and his own construction of nodec group topologies, and proves two $$P$$-point theorems. By a $$P$$-point theorem, he means any statement of the form “If there exists a topological group with some properties then there is a $$P$$-point in $$\omega^*$$”. The first $$P$$-point theorem was proved by the reviewer for a maximal topological group. Theorem 5.19 is a generalization of this result. Theorem 5.14 is a $$P$$-point theorem for extremally disconnected topological groups containing a countable discrete non-closed subset.
Chapter 6. The semigroup $$\beta S$$. This is a background.
Chapter 7. Ultrafilter semigroups. For a semigroup $$S$$ with identity $$1$$, endowed with a left invariant $$T_1$$ topology $$\tau$$, $$\text{Ult}(\tau)$$ denotes the subsemigroup of $$\beta S_{\text{discrete}}$$ of all ultrafilters converging in $$\tau$$ to $$1$$. Some basic facts about $$\text{Ult}(\tau)$$ and interrelations between $$\text{Ult}(\tau)$$ and $$(S,\tau)$$, which appeared at the very dawn of the theory, are collected here. Theorem 7.24 gives some conditions on a compact right topological semigroup $$T$$ under which $$T$$ is a continuous homomorphic image of the canonical semigroup $$\mathbb{H}_\kappa$$. Theorem 7.29 shows when and how $$\text{Ult}(\tau)$$ can be mapped onto $$\beta\mathbb{N}$$.
Chapter 8. Finite groups in $$\beta G$$. A bit of reminiscences. One nice September day in 1976, I asked Yevgen if any two countable topological groups $$G$$ and $$H$$ of countable weight are locally isomorphic, i.e., there is a homeomorphism $$f:G\to H$$ such that, for each $$x\in G$$, one can find a neighbourhood $$U$$ of the identity of $$G$$ such that $$f(xy)=f(x)f(y)$$ for every $$y\in U$$. A week later he brought a seminal version of Theorem 8.9 answering this question in the affirmative. My advice as a supervisor was to look around for possible applications. A week or two later he proved that each finite group in $$\beta\mathbb{Z}$$ is a singleton. It was a great surprise! Analyzing his arguments, Dona Strauss proved Theorem 8.18. At last, in the frame of Zelenyuk’s technique, I proved Theorem 8.33 characterizing all finite subgroups in $$\beta S$$ for a countable group $$S$$.
Chapter 9. Ideal structure of $$\beta G$$. For an infinite group $$G$$ of cardinality $$\kappa$$, we see some decompositions of the ideal $$U(G)$$ of all uniform ultrafilters into “small” closed left ideals, $$2^{2^\kappa}$$ minimal right ideals, and a free group of rank $$2^{2^\kappa}$$ in the minimal right ideal of $$\beta G$$ if $$G$$ is Abelian.
Chapter 10. Almost maximal topological group. This very delicate chapter is about a topological group $$(G,\tau)$$ with finite semigroup $$\text{Ult}(\tau)$$. If $$G$$ is countable then $$\text{Ult}(\tau)$$ must be a projective in the category $$\mathcal{F}$$ of all finite groups. On the other hand, under $$\mathfrak{p}=\mathfrak{c}$$, for every projective $$F$$ in $$\mathcal{F}$$, there exists a group topology $$\tau$$ on the countable Boolean group $$G$$ such that $$\text{Ult}(\tau)$$ is isomorphic to $$F$$. The class of countable almost maximal topological groups satisfies the $$P$$-point theorem. The remainder is about projectives in $$\mathcal{F}$$. I am curious why the author did not give a proof of the final Theorem 10.41, but refers us to [95] of the references.
Chapter 11. Almost maximal spaces. In fact, this is about left topological groups $$(G,\tau)$$ with finite semigroup $$\text{Ult}(\tau)$$. The author develops some transfinite local technique and proves, in particular, that every infinite group $$G$$ admits a zero-dimensional maximal topology of dispersion character $$|G|$$.
Chapter 12. Resolvability. This chapter corresponds with Chapter 5. By Theorem 12.6 every countable non-discrete topological group with no elements of order 2 can be partitioned into $$\omega$$ dense subsets. The proof is based on a “canonization” of homeomorphisms of finite order. Theorem 12.13 is a $$P$$-point theorem for countable $$\omega$$-irresolvable topological groups. By Theorem 12.20, every Abelian group with no elements of order 2 can be partitioned into $$\omega$$ subsets dense in each group topology on $$G$$.
Chapter 13. Open problems. This section contains 18 open questions with no comments.
Reviewer’s remark: The author is very specific in the choice of material, citations and comments. For a slightly different view on the subject see the survey [I. V. Protasov, Algebra Discrete Math. 2009, No. 1, 83–110 (2009; Zbl 1199.22002)] and the book [M. Filali and I. Protasov, Ultrafilters and Topologies on Groups, Lviv: VNTL Publisher (2011)]. Nevertheless, the book under review is an interesting book.

##### MSC:
 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 22A05 Structure of general topological groups 22A15 Structure of topological semigroups 54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
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