The Addition theorem for two-step nilpotent torsion groups

Shlossberg, Menachem

doi:10.48550/arxiv.2201.12860

Cited by 1 publication

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“…On the other hand, the first instance of (at alg ) for N-actions on torsion Abelian groups was given in [18], while the general case for N-actions on Abelian groups was settled in [16]. Moreover, (at alg ) holds also for N-actions on some special classes of non-Abelian groups [25,26,53].…”

Section: Introductionmentioning

confidence: 99%

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem

Dikranjan¹,

Bruno²,

Virili³

2023

Preprint

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For a left action S λ X of a cancellative right amenable monoid S on a discrete Abelian group X, we construct its Ore localization G λ * X * , where G is the group of left fractions of S ; analogously, for a right action K ρ S on a compact space K, we construct its Ore colocalization K * ρ * G. Both constructions preserve entropy, i.e., for the algebraic entropy h alg and for the topological entropy h top one has h alg (λ) = h alg (λ * ) and h top (ρ) = h top (ρ * ), respectively.Exploiting these constructions and the theory of quasi-tilings, we extend the Addition Theorem for h top , known for right actions of countable amenable groups on compact metrizable groups [36], to right actions K ρ S of cancellative right amenable monoids S (with no restrictions on the cardinality) on arbitrary compact groups K.When the compact group K is Abelian, we prove that h top (ρ) coincides with h alg (ρ ∧ ), where S ρ ∧ X is the dual left action on the discrete Pontryagin dual X = K ∧ , that is, a so-called Bridge Theorem. From the Addition Theorem for h top and the Bridge Theorem, we obtain an Addition Theorem for h alg for left actions S λ X on discrete Abelian groups, so far known only under the hypotheses that either X is torsion [10] or S is locally monotileable [11].The proofs substantially use the unified approach towards entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids, as developed in [17] (for N-actions) and [58].

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