On a Compactness Property of Topological Groups

For a domain  in C, we denote by Aut() the group of holomorphic automorphisms of . It is obvious that Aut() is a topological group with respect to the law of composition and the compact-open topology. In particular, it is a theorem of H. Cartan that Aut() is in fact a Lie group if  is bounded. In light of the outstanding question “Which domains possess noncompact automorphism group?” there is much interest focused upon the existence and nonexistence of orbits of the automorphism group action accumulating at a given boundary point. The well-known Greene–Krantz conjecture belongs to such a line of research. In this paper, we discuss the finite-type boundary points that repel automorphism orbits. Denote by τ6(q) the D’Angelo type (see [10]) at q of the real hypersurface 6 in C. Now we present our main theorem.

Let F be a discrete uniform subgroup of a connected simply connected solvable Lie group S. It is shown how S is essentially determined by F, using the point of view of group extensions. Let r be a discrete uniform subgroup of a connected simply connected solvable Lie group S. The main purpose of this work is to study how S is essentially determined by T from the point of view of group extensions. In [5], Mal'cev proved that T determines 5 uniquely if S is nilpotent. However, for a general solvable group, the situation seems less favorable. Nevertheless, L. Auslander (Theorem 2 in [1]) has obtained some results when T is the fundamental group of a nilmanifold, and in the subsequent works of Auslander and Tolimieri ([2], [8], [9]), stronger results were proved by using the notation of semisimple splittings, from which one can obtain the generalization of the result in [1] to arbitrary discrete uniform subgroups T. It is the purpose of this paper to revisit and prove directly this generalization by making use of the theory of group extensions as Auslander originally did in [1]. The author is indebted to the referee for pointing out the above results of Auslander and Tolimieri. 1. Let G and H be topological groups. An extension of G by H is a pair (E, rr) consisting of a topological group E which contains G as a closed normal subgroup and a continuous open homomorphism it of E onto H whose kernel is G. Two extensions (Et, 7r,-) of G by H are said to be equivalent if there exists an isomorphism a : Ex-+E2 of topological groups which leaves elements of G fixed and is such that tt2o=ttx. If (E, it) is an extension of G by H, then this determines a homomorphism Tr°:H—>0(G), where 0(G) denotes the group A(G) of all automorphisms of G modulo the inner automorphism group 1(G) of G. If (Ex, irx) and (E2, tt2) are equivalent, then 77^=712. For any homomorphism -0(G), let Ext(G, H, cp) denote the set of all equivalence classes of the extensions (E, it) of G by H Received by the editors July 12, 1971 and, in revised form, November 17, 1971. AMS 1970 subject classifications. Primary 22E15, 22E20, 22E40.

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the invariant cuts of the model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is replaced by $\boldsymbol{e}$. First of all, there is no written proof in the available literature that the group over $\boldsymbol{e}$ is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other hand, we extend Newelski's theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context.

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

All Hartman–Mycielski groups are minimally almost periodic

Sequential criteria for equicontinuity and uniformities on topological groups

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.

A systematic method is presented whereby any compact Lie group of n-real parameters is dealt with from an infinitesimal approach with the representative matrix method based on a group of inner automorphisms suggested in a previous paper. The group manifold, defined in terms of a metric of group parameters, is identified as a Riemannian one in which these parameters play a role of n curvilinear coordinates. Riemannian geometry is thus valid in the group manifold, and geometric quantities are explicitly calculated in terms of the symmetric or (0) connection by a straightforward application of the ordinary procedure of tensor analysis. A new and simpler method of computing the invariant volume element is presented within this framework. Furthermore, we discuss in detail the group of inner automorphisms for the calculation of the matrix element of finite rotations for any irreducible representation (abbreviated MEFRIR). It is found that our method works very well and yields right and left vector fields together with a set of 2n equations to be satisfied by the MEFRIR. The global properties of the group may, therefore, be obtained as a solution to these equations. Moreover, it provides not only the generalized Maurer-Cartan equations, the Lie structure formulas, two parameter groups of point transformations and the adjoint group, but also two additional nonsymmetric (+) and (−) connections with zero curvature, which do not possess any preassigned metric but possess two absolute parallelisms. Thus, our results on differential geometry completely agree with Cartan and Schouten's. A link between differential geometry and representations is presented by the right and left vector fields which are explicitly calculable in terms of the n parameters and through which geometric quantities, e.g., the Riemann tensor, the Ricci tensor, and the scalar curvature, of any connection are explicitly displayed. A theorem relating both the vector fields to the metric tensors is also included. Finally, the l (rank of the group) invariant differential equations to be satisfied by the MEFRIR are cast in the covariant (or Lie derivative) forms in any connection. Examples of the invariant equations are given for SU(2), SO(3), and SU(3). The two invariant equations of the latter can be cast in terms of the eigenvalues of isospin and hypercharge upon carrying out charge and hypercharge quantizations; in this connection, a new nonrelativistic wave equation to be satisfied by SU(3) multiplets as a generalization of the Schrödinger equation of the symmetric top is also proposed.

For a locally compact group $G$, the dual space $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $\hat G$ in the case that $G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $\hat G$; (2) the “discrete series” of square-integrable representations does in fact inherit the discrete topology from $\hat G$; (3) the topology of the reduced dual ${\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.

Product graphs for given subgroups of the wreath product of two groups, II

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.