Precompact groups and convergence

Convergent sequences in iterated ultrapowers as p-compact groups

Totally Lindelöf and totally ω-narrow paratopological groups

A mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

We prove that there exists a countable metrizable topological group G G such that every countable metrizable group is isomorphic to a quotient of G G . The completion H H of G G is a Polish group such that every Polish group is isomorphic to a quotient of H H .

We investigate properties of the clopen type semigroup of an action of a countable group on a compact, 0 -dimensional, Hausdorff space X . We discuss some characterizations of dynamical comparison (most of which were already known in the metrizable case) in this setting and prove that for a Cantor minimal action \alpha of an amenable group the topological full group of \alpha admits a dense, locally finite subgroup iff the corresponding clopen type semigroup is unperforated. We also discuss some properties of clopen type semigroups of the Stone–Čech compactifications and universal minimal flows of countable groups, and derive some consequences on generic properties in the space of minimal actions of a given countable group on the Cantor space.

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.

Fixed points for bounded orbits in Hilbert spaces

A now standard technique for treating a class of Baire set problems, originating with Halmos [2; 51.D], is to construct a metric space that, in a way, "fixes" a given Baire set. Formally this means, given a Baire set B in a space X, find a metric space M and a map g of X onto M with the property that B = g l gB . In phrasing the problem for topological groups it is natural to require that M be a group and g be a homomorphism. Specifically, what is desired is a subgroup H that "fixes" B, i.e., such that B = BH. This problem is solved in Halmos for locally compact a-compact groups in a theorem [2; 64.G] that has led to interesting consequences elsewhere [4, 1]. As will be seen, Halmos' theorem is also true for totally bounded groups. An example of Sirota's of a countable topological group with no non-trivial metrizable homomorphic images indicates that, in this direction at least, the barrier of local compactness cannot be lifted without giving up metrizability in M. If this is done, however, replacing the requirement that M be metrizable, i.e., that H have a countable base in G, with the less stringent requirement that H be a G~-set in G, an analogue of Halmos' theorem is true for Lindelff groups. In view of these remarks it is convenient to distinguish between two distinct Baire set fixing problems. A closed invariant subgroup H of a group G is said to be a fixiru3 subgroup (M-fixing subgroup) provided H is a G~-set (H has a countable base of neighborhoods in G). G satisfies the Baire set fixing property (M-Baire set fixing property) provided, for each Baire set in G, there is a fixing subgroup (M-fixing subgroup) such that B = BH. The Baire sets are taken to be the elements of the smallest a-algebra containing the zero sets of continuous real-valued functions on G. For technical convenience in describing proofs, a G~-set containing the identity is called an admissible set.

The notion of statistical convergence of double sequences in topological groups is introduced by Çakalli and Savaş. The goal of this paper is to introduce a class of summability methods that can be applied to \(\lambda \)- double statistical convergence in topological groups and is to prove some theorems.Keywords\(\lambda \)- double statistical convergenceTopological groupsPringsheim limit

Recently, Patterson and Savas (Math. Commun. 10:55-61, 2005), defined the lacunary statistical analog for double sequences as follows: A real double sequences is said to be P-lacunary statistically convergent to L provided that for each , . In this case write or . In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems. MSC:42B15, 40C05.

Thin subsets of topological groups

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H.As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.In [9], G. W. Mackey gave an argument using Haar measures to prove the following: Assume G is locally compact topological group.If H is a subgroup such that the space G/H, formed by giving the set of (left) H-cosets the quotient Borel structure, is countably separated, then H is closed in G.We will show that this result can be obtained by an analogous argument using the theory of Baire category.The category version has a wider application and shows that the above statement remains true under the weaker assumption that G is totally nonmeager.In particular, it holds whenever G is topologically complete (in the sense of Cech).As a corollary, we will prove that a well-known theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish (separable, completely metrizable) groups holds for Borel actions as well.We will also show that a recent result of R. Vaught on decompositions of invariant analytic sets has a similar extension to Borel actions.Each step in our main argument corresponds to an analogous step in Mackey's proof.The reader may find it interesting to compare our development with the proof of Theorem 7.2 in Mackey [9].Recall (cf.[2, IX,) that a topological group G which is not meager (first category) is a Baire space (i.e.no nonempty open subset of G is meager in G).G is totally nonmeager if no nonempty closed subset of G is meager in itself.Every topologically complete space, in particular every locally compact space and every completely metrizable space, is totally nonmeager (cf.[7, III, 8]).B E G is strictly Baire (or has the restricted Baire property)

Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.

I. V. M. Klimkin, "On extension of a vector measure," Izv. Vyssh. Uchebn. Zaved., Mat., ~, 46-53 (1971). 2. V. M. Klimkin, "On certain properties of continuous outer measures," Uch. Zap. Leningr. Gos. Pedagog. Inst., 469, 187-204 (1970). 3. N. S. Gusel'nikov, "On the convergence of a sequence of continuous triangular measures," Nauch. Dokl., Mat., 26, 78-82, Hercen Lectures, Izd. Mosk. 0blast. Pedagog. Inst. im N. K. Krypskoi, Moscow (1973). 4. N. S. Gusel'nikov, "On the extension of quasi-Lipschitz set functions," Mat. Zametki, 17, No. i, 21-31 (1975). 5. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Wiley, New York (1958). 6. J. L. Kelley, General Topology, Van Nostrand, New York (2955). 7. L. Ya. Savel'ev, "On the extension of a mappin E of a Boolean ring into a topological Abelian group," Dokl. Akad. Nauk SSSR, 149, No. 6, 1268-1269 (1963). 8. L. Ya. Savel'ev, "Extension of measures by continuity," Sib. Mat. Zh., ~, No. 3, 639650 (1964). 9. L. Ya. Savel'ev, "Certain conditions for extendability of vector measures," Sib. Mat. Zh., 9, No. 4, 937-950 (1968). 10. M. Kh. Khafizov, "On the extension of a vector-valued measure," in: Current Problems of Mathematical Logic and Set Theory [in Russian], Izd. Mosk. Gos. Pedagog. Inst., Moscow (1975), pp. 291-301. ii. P. R. Halmos, Measure Theory, Van Nostrand, New York (1950).

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every unbounded tower set of reals X of cardinality aleph_1, the space Cp(X) is productively FU. In particular, the set X is productively gamma. 2. Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming CH, there are gamma spaces whose product is not even Menger. 3. Solving a problem of Scheepers-Tall, we show that the properties gamma and Gerlits--Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that Remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel'skii duality to obtain results concerning local properties of function spaces and countable topological groups.