Volume 8, Issue 1

We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.

If H is a subgroup of an abelian group G and φ ∈ End( G ), H is called φ-inert (and φ is H-inertial ) if φ ( H ) ∩ H has finite index in the image φ ( H ). The notion of φ -inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ , while inertial endo-morphisms (these are endomorphisms that are H -inertial for every subgroup H ) were intensively studied by Rinauro and the first named author. A subgroup H of an abelian group G is said to be fully inert if it is φ -inert for every φ ∈ End( G ). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G . It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p -groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |( H + φH )/ H | ≤ n for every φ ∈ End( G ). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C * -algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives rise to examples of C * -algebras (1.9) generalizing the boundary quotients ∂Cλ*(𝒨)\partial C_\lambda ^*(\mathcal{M}) discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients, while the cone topology can be used to define canonical measures on the attractor of an ℳ-fractal for a finitely 1-generated monoid ℳ.

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

Let 𝕂 be a discrete field and ( V , ϕ ) a pair consisting of a locally linearly compact 𝕂-space V and a continuous endomorphism ϕ : V → V . We provide the formulae to compute the topological entropy ent * of the flow ( V , ϕ ) subject to either extension or restriction of scalars.

We construct a metrizable semitopological semilattice X whose partial order P = {( x , y ) ∈ X × X : xy = x } is a non-closed dense subset of X × X . As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with ( xV ∪ Vx ) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

The set 𝒜 := 𝔺δ 0 + 𝒟 + ′, obtained by attaching the identity δ 0 to the set 𝒟 + ′ of all distributions on 𝕉 with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equivalently, every tall left-invertible matrix with entries from 𝒜 can be completed to a square matrix with entries from 𝒜, which is invertible.