Hausdorff Group Research Articles

Rigid topologies on groups

Our main result is to show that every infinite, countable, residually finite group G admits a Hausdorff group topology which is neither discrete nor precompact.

On normal subgroups in automorphism groups

Abstract 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.

Noncommutative numerable principal bundles from group actions on C⁎-algebras

We introduce the notion of a locally trivial G-C⁎-algebra, which is a noncommutative counterpart of the total space of a locally compact Hausdorff numerable principal G-bundle. To obtain this generalization, we have to go beyond the Gelfand–Naimark duality and use the multipliers of the Pedersen ideal. Our new concept enables us to investigate local triviality of noncommutative principal bundles coming from group actions on non-unital C⁎-algebras, which we illustrate through examples coming from C0(Y)-algebras and graph C⁎-algebras. In the case of an action of a compact Hausdorff group on a unital C⁎-algebra, local triviality in our sense is implied by the finiteness of the local-triviality dimension of the action. Furthermore, we prove that if A is a locally trivial G-C⁎-algebra, then the G-action on A is free in a certain sense, which in many cases coincides with the known notions of freeness due to Rieffel and Ellwood.

Weak and strong type inequalities for Orlicz spaces

We revisit the generalisation of Calderon’s Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of σ-compact locally compact Hausdorff groups acting measure-preservingly on σ-finite measure spaces. In particular we develop a much more robust protocol for transferring weak and strong type inequalities from Orlicz spaces in the group setting to Orlicz spaces in the measure space setting. This is an important addition to the protocol developed in [7], which to date has only yielded weak type inequalities. The current approach also places fewer restrictions on the underlying Young functions describing the Orlicz spaces involved.

On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of [Formula: see text]. Using Poincaré series of [Formula: see text]-finite matrix coefficients of integrable discrete series representations of [Formula: see text], we construct a spanning set for the space [Formula: see text] of Siegel cusp forms of weight [Formula: see text]. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of [Formula: see text], we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of [Formula: see text]. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

On compact groups with Engel-like conditions

Let G be a compact Hausdorff group. We show that if for every element there exists a positive integer such that xq is Engel, then G is locally virtually nilpotent. Furthermore, we show that if G is a finitely generated compact Hausdorff group in which any commutator in G is Engel, then the commutator subgroup is locally nilpotent.

Products of locally minimal groups

A topological group G is called locally minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f:G→H is a continuous isomomorphism such that f(V) is a neighbourhood of the identity in H, then f is open. This paper is focused on the study of products of locally minimal groups. Local minimality is not preserved under taking products. We call a topological groups G perfectly locally minimal group if G×H is locally minimal for every locally minimal group H. The well known criterion of Stoyanov for perfectly minimal abelian groups is extended, first to the case of arbitrary perfectly minimal groups. Then its counterpart for perfectly locally minimal groups is provided. In particular, we show that a topological group G is perfectly locally minimal if and only if G×(Z,τp) is locally minimal for every prime p, where τp denotes the p-adic topology on Z. Complete locally minimal groups turn out to be perfectly locally minimal. This motivates a special attention towards (local) minimality of complete groups. A contribution is given towards Uspenksij's problem on minimality of complete groups, by proving that if {Gi:i∈I} is a family of minimal groups with minimal quotient Gi/Z(Gi) the product G=∏i∈IGi is minimal if and only if Z(G) is minimal. Furthermore, we answer a question from [14] about closed subgroups of a product of locally compact abelian groups, showing that they are locally compact if and only if they are locally minimal. Finally, a general criterion for local minimality of arbitrary products is given based on already known criteria for minimality of arbitrary products [10].

Topologically stable and persistent points of group actions

In this paper, we introduce topologically stable points, persistent points, persistent property, persistent measures and almost persistent measures for first countable Hausdorff group actions of compact metric spaces. We prove that the set of all persistent points is measurable and it is closed if the action is equicontinuous. We also prove that the set of all persistent measures is a convex set and every almost persistent measure is a persistent measure. Finally, we prove that every equicontinuous pointwise topologically stable first countable Hausdorff group action of a compact metric space is persistent. In particular, every equicontinuous pointwise topologically stable flow is persistent.

On pp-elimination and stability in a continuous setting

We generalize pp-elimination for modules, or more generally, abelian structures, to a continuous logic setting where the abelian structure is equipped with a homomorphism to a compact Hausdorff group. We conclude that the continuous logic theory of such a structure is stable.

Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups

For a locally compact Hausdorff group G and the compact space \mathrm{Sub}_G of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on \mathrm{Sub}_G in terms of distality and expansivity. We prove that an infinite discrete group G , which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \mathrm{Sub}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . For an automorphism T of a connected Lie group G which keeps a lattice \Gamma invariant, we compare the behaviour of the actions of T on \mathrm{Sub}_G and \mathrm{Sub}_\Gamma in terms of distality. Under certain necessary conditions on the Lie group G , we show that T acts distally on \mathrm{Sub}_G if and only if it acts distally on \mathrm{Sub}_\Gamma . We also obtain certain results about the structure of lattices in a connected Lie group.

Integral of an extension of the sine addition formula

In this paper, we determine the continuous solutions of the integral functional equation of Stetkær's extension of the sine addition law ∫Gf(xyt)dμ(t)=f(x)χ1(y)+χ2(x)f(y), x,y∈G, where f:G→C, G is a locally compact Hausdorff group, μ is a regular, compactly supported, complex-valued Borel measure on G and χ1, χ2 are fixed characters on G.

Fourier transform on compact Hausdorff groups

This article deals with the generalization of the abstract Fourier analysis on the compact Hausdorff group. In this paper, the generalized Fourier transform F is defined as F (?)(?) = R ?(h)M? (h?1) d? (h) for all ? ? L2 (G) ? L1 (G), where M? is a continuous unitary representation M? : G ? UC (Cn(?)) of the group G in Cn(?), and its properties are studied. Also, we define the symplectic Fourier transform and the generalized Wigner function WA (?, ?) and establish the Moyal equality for the Wigner function. We show that the homomorphism ? : G ? U (L2 (G/K,H1)) induced by ? : G ? (G/K) ? U(H1) by (? (?)) (g, h) = (? (h?1, g))?1 (? (h?1g)), g ? G/K, h ? G, ? ? L2 (G/K,H1) is a unitary representation of the group G, assuming the mapping h 7? (?(?)) (g, h) is continuous as morphism G ? U (L2 (G/K,H1)). We study the unitary representation ?? : G ? H induced by the unitary representation V : K ? U(H1) given by ??g (?) (t) = ? (g?1t) for all t ? G/K.

Abstract group actions of locally compact groups on CAT(0) spaces

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris–Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion-free CAT(0) group is continuous.

Automatic continuity for groups whose torsion subgroups are small

Abstract We prove that a group homomorphism φ : L → G \varphi\colon L\to G from a locally compact Hausdorff group 𝐿 into a discrete group 𝐺 either is continuous, or there exists a normal open subgroup N ⊆ L N\subseteq L such that φ ⁢ ( N ) \varphi(N) is a torsion group provided that 𝐺 does not include ℚ or the 𝑝-adic integers Z p \mathbb{Z}_{p} or the Prüfer 𝑝-group Z ⁢ ( p ∞ ) \mathbb{Z}(p^{\infty}) for any prime 𝑝 as a subgroup, and if the torsion subgroups of 𝐺 are small in the sense that any torsion subgroup of 𝐺 is artinian. In particular, if 𝜑 is surjective and 𝐺 additionally does not have non-trivial normal torsion subgroups, then 𝜑 is continuous. As an application, we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.

The inner kernel theorem for a certain Segal algebra

The Segal algebra {mathbf{S}}_{0}(G) is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups G. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from {mathbf{S}}_{0}(G_1) to {mathbf{S}}'_{0}(G_2) by generalized functions in {mathbf{S}}'_{0}(G_1 times G_2) the “outer kernel theorem”. The main subject of this paper is to formulate what we call the “inner kernel theorem”. This is the characterization of those linear operators that have kernels in {mathbf{S}}_{0}(G_1 times G_2). Such operators are regularizing—in the sense that they map {mathbf{S}}'_{0}(G_1) into {mathbf{S}}_{0}(G_2) in a w^{*} to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies “integrate” to a Dirac delta distribution in a mathematically justifiable way.

Transversality on locally pseudocompact groups

Two non-discrete Hausdorff group topologies τ and δ on a group G are called transversal if the least upper bound τ ⋁ δ of τ and δ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affirmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153: 3338–3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.

Approximations in L^1 with convergent Fourier series

For a separable finite diffuse measure space {mathcal {M}} and an orthonormal basis {varphi _n} of L^2({mathcal {M}}) consisting of bounded functions varphi _nin L^infty ({mathcal {M}}), we find a measurable subset Esubset {mathcal {M}} of arbitrarily small complement |{mathcal {M}}{setminus } E|<epsilon , such that every measurable function fin L^1({mathcal {M}}) has an approximant gin L^1({mathcal {M}}) with g=f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of {mathcal {M}}=G/H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Characterizations of self-adjointness, normality of pseudo-differential operators on homogeneous space of compact groups

Let G be a compact Hausdorff group and H be a closed subgroup of G. In this paper, we show that every bounded linear operator T on is a pseudo-differential operator with the symbol σ for . We present necessary and sufficient conditions on symbols to ensure that a bounded pseudo-differential operator on is self-adjoint, normal and present explicit formula for their symbols. A necessary and sufficient condition is also given such that the bounded linear operators on posses eigenvalues and eigenfunctions.

Forcing a classification of non-torsion Abelian groups of size at most [formula omitted] with non-trivial convergent sequences

We force a classification of all the Abelian groups of cardinality at most 2c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2c, by showing that if a non-torsion Abelian group of size at most 2c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

On Sjölin–Soria–Antonov type extrapolation for locally compact groups and a.e. convergence of Vilenkin–Fourier series

A Sjolin–Soria–Antonov type extrapolation theorem for locally compact $$\sigma$$ -compact non-discrete Hausdorff groups is proved. Applying this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class $$L {\rm log}^{+} L {\rm log}^{+} {\rm log}^{+} {\rm log}^{+} L$$ .