Final group topologies,Kac-Moody groups and Pontryagin duality

We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k _ω-space, or locally k _ω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of k _ω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally k _ω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally k _ω abelian groups.     

Topological groups and recurrence of quasi transitive graphs

We present all steps which are necessary in order to classify all locally finite, infinite graphs which carry a quasi transitive random walk that is recurrent. Some new and/or simpler proofs are given. Most of them rely on the fact that autmomorphism groups of locally finite graphs are locally compact with respect to the topology of pointwise convergence—this allows the use of integration on these groups. Conferenza tenuta il 28 novembre 1994     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

Groupoid cohomology and extensions

We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.

     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Duality between compactness and discreteness beyond pontryagin duality

One of the most striking results of Pontryagin’s duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height 2 whose dual spaces are quasicompact and non-Hausdorff (they are T ₁ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a T ₁ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.     

Locally soluble infinite-dimensional linear groups with restrictions on nonAbelian subgroups of infinite ranks

We are concerned with locally soluble linear groups of infinite central dimension and infinite sectional p-rank, p ⩾ 0, in which every proper non-Abelian subgroup of infinite sectional p-rank has finite central dimension. It is proved that such groups are soluble. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 601–616, September–October, 2008.     

On infinite groups with a given strongly isolated 2-subgroup

It is proved that some groups with a strongly isolated 2-subgroup of period not exceeding four are locally finite. In particular, the positive answer to Shunkov’s question 10.76 in the Kourovka notebook is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 272–285, August, 2000.     

Pseudocharacters on anomalous products of locally indicable groups

The question on the existence of nontrivial pseudocharacters on anomalous products of locally indicable groups is considered. Some generalizations of theorems of R. I. Grigorchuk and V. G. Bardakov on the existence of nontrivial pseudocharacters on free products with the amalgamation subgroup are found. It is proved that they exist on an anomalous product 〈G, x | w = 1〉, where G is a locally indicable noncyclic group. We also prove some other propositions on the existence of nontrivial pseudocharacters on anomalous products of groups. Results on the second cohomologies of these products and their nonamenability follow from the propositions on the existence of nontrivial pseudocharacters on these groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 55–64, 2006.     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.     

Locally compact subgroups of the spectrum of the measure algebra II

The maximal ideal space Δ_G of the measure algebra M(G) of a locally compact abelian group G is a compact commutative semitopological semigroup. In this paper we show that cℓ Ĝ the closure of Ĝ, the dual of G, in Δ_G can contain maximal subgroups which are not locally compact. We have previously characterized the locally compact maximal subgroups of cℓ Ĝ as arising from locally compact topologies on G which are finer than the original topology. This research was supported in part by NSF contract number GP-19852.     

On a theorem of Wiener

 Wiener has shown that an integrable function on the circle T which is square integrable near the identity and has nonnegative Fourier transform, is square integrable on all of T. In the last 30 years this has been extended by the work of various authors step by step. The latest result states that, in a suitable reformulation, Wiener's theorem with ``p-integrable' in place of ``square integrable' holds for all even p and fails for all other p  (1, ∞) in the case of a general locally compact abelian group. We extend this to all IN-groups (locally compact groups with at least one invariant compact neighbourhood) and show that an extension to all locally compact groups is not possible: Wiener's theorem fails for all p < ∞ in the case of the ax + b-group. Received: 12 September 2000 Mathematics Subject Classification (2000): 43A35     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.     

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact p-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a “generalized Robba ring” for uniform pro-p groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a selfdual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.     

Continuous Selections and Locally Pseudocompact Groups

We characterize locally pseudocompact groups by means of the selection theory. Our result is the selection version of the well-known Comfort—Ross theorem on pseudocompactness which states that a topological group is pseudocompact if and only its Stone—Čech compactification is a topological group.     

Normal ergodic actions and extensions

We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum.     

Partial Densities on Locally Compact Abelian Groups and Uniformly Distributed Sequences

 We investigate conditions under which a partial density on a locally compact abelian group can be extended to a density. The results allow applications to the theory of uniform distribution of sequences in locally compact abelian groups. Received August 27, 2001 Published online July 12, 2002