Intrinsic algebraic entropy

Dikranjan, Dikran; Bruno, Anna Giordano; Salce, Luigi; Virili, Simone

doi:10.1016/j.jpaa.2014.09.033

Cited by 33 publications

(55 citation statements)

References 21 publications

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“…Indeed, Corollary 2 and a limit-free formula as in [4,15] provide ent dim (β K ) = dim K (V − /β −1 K (V − )) = 1. Quite remarkably, φ-inert subspaces do not enrich the dynamics of linear flows like φ-inert subgroups do in the framework of abelian groups (see [9]). Indeed, one verifies that ent dim (φ) = ent dim (φ) for every φ : V → V , where ent dim (φ) := sup lim n→∞ dim K (T n (φ, F )) n | F ≤ V and dim K (F ) < ∞ , which is a classical entropy function for vector spaces and their endomorphisms (details about this entropy function can be found in [3]).…”

Section: Connection With Algebraic Entropymentioning

confidence: 99%

A property of the lamplighter group

Castellano

Cook

Kropholler

2020

Topological Algebra and Its Applications

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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.A subgroup H of a group G is said to be inert if H and g −1 Hg are commensurate for all g ∈ G, meaning that H ∩ H g always has finite index in both H and H g . The terminology was introduced by Kegel and has been explored in many contexts (see, for example, the recent survey [8]). In abstract group theory, Robinson's investigation [13] focusses on soluble groups. Here we study a particular special family of soluble groups: the lamplighter groups and our interest is in the connection with the theory of totally disconnected locally compact groups. In that context, inert subgroups are particularly important in the light of van Dantzig's theorem that every totally disconnected locally compact group has a compact open subgroup and of course all such subgroups are commensurate with one another and therefore inert. It should be noted that in recent literature it is common to use the term commensurated in place of inert, see for example [5,6,7,10]. In §2 we remark a dynamical aspect of the property investigated here and relate it to the concept of algebraic entropy.The relation of commensurability is an equivalence relation amongst the subgroups of a group. By a class we shall here mean an equivalence class of subgroups under this relation. For a prime p, the corresponding lamplighter group is the standard restricted wreath product F p wr Z, i.e., the standard restricted wreath product of a group of order p by an infinite cyclic group. These are the simplest of soluble groups that fall outside the classes considered by Robinson [13]. Our main observation is as follows.Theorem. The inert subgroups of the lamplighter group F p wr Z fall into exactly five classes.

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Section: Connection With Algebraic Entropymentioning

confidence: 99%

A property of the lamplighter group

Castellano

Cook

Kropholler

2020

Topological Algebra and Its Applications

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“…We start with some results already proved in [6], or easily deducible from them. We provide their essential details, for the sake of completeness.…”

Section: -Inert and Fully Inert Subgroupsmentioning

confidence: 99%

“…In Section 2 we collect preliminary results on -inert subgroups, some of which are extracted from [6], and on fully inert subgroups. In Section 3 fully inert subgroups of divisible groups are investigated, as a preparation for the main results of the paper contained in Sections 4, 5 and 6, where we characterize inert groups as follows.…”

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confidence: 99%

Fully inert subgroups of divisible Abelian groups

Dikranjan

Bruno

Salce

et al. 2013

Journal of Group Theory

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Abstract.A subgroup H of an Abelian group G is said to be fully inert if the quotient .H C .H //=H is finite for every endomorphism of G. Clearly, this is a common generalization of the notions of fully invariant, finite and finite-index subgroups. We investigate the fully inert subgroups of divisible Abelian groups, and in particular, those Abelian groups that are fully inert in their divisible hull, called inert groups. We prove that the inert torsion-free groups coincide with the completely decomposable homogeneous groups of finite rank and we give a complete description of the inert groups in the general case. This yields a characterization of the fully inert subgroups of divisible Abelian groups.

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“…As in [6] and [7], an endomorphism ϕ of an abelian group A (from now on always in additive notation) is said (right-) inertial iff:…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, notice that in [7] the concept of inertial endomorphism is related to the investigation of the dynamical properties of an endomorphism of an abelian group.…”

Section: Introductionmentioning

confidence: 99%

Inertial endomorphisms of an abelian group

Dardano

Rinauro

2014

Annali di Matematica

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We describe inertial endomorphisms of an abelian group A, that is endomorphisms\ud ϕ with the property |(ϕ(X) + X)/X| < ∞ for each X ≤ A. They form a ring\ud I E(A) containing the ideal F(A) formed by the so-called finitary endomorphisms, the ring\ud of power endomorphisms and also other non-trivial instances.We show that the quotient ring\ud I E(A)/F(A) is commutative. Moreover, inertial invertible endomorphisms form a group,\ud provided A has finite torsion-free rank. In any case, the group I Aut(A) they generate is\ud commutative modulo the group FAut (A) of finitary automorphisms, which is known to be\ud locally finite. We deduce that I Aut(A) is locally-(center-by-finite). Also, we consider the\ud lattice dual property, that is |X/(X ∩ϕ(X))| < ∞for each X ≤ A and show that this implies\ud the above one, provided A has finite torsion-free rank

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Intrinsic algebraic entropy

Cited by 33 publications

References 21 publications

A property of the lamplighter group

A property of the lamplighter group

Fully inert subgroups of divisible Abelian groups

Inertial endomorphisms of an abelian group

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