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.

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.

Reviewer: Igor Vladimirov Protasov (Kyiv)

##### 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.) |