Entropy on modules over the group ring of a sofic group

Liang, Bingbing

doi:10.1090/proc/14271

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…Define [18, Definition 3.1]

dim false(M_{1} false| M_{2} false) : = \underset{A \in F ({scriptM}_{1})}{trueprefixsup} \underset{B \in F ({scriptM}_{2})}{trueprefixinf} \underset{F \in F (Γ)}{trueprefixinf} \lim_{j \to ω} \frac{L (M false(scriptA, scriptB, F, σ_{j} false))}{false| X_{j} false|} .

Then

dim (\cdot | \cdot)

is a bivariant Sylvester module rank function for

R Γ

[18, Theorem 1.1, Corollary 3.2, Proposition 3.4, Proposition 3.5]. For connections of this bivariant Sylvester module rank function to dynamical invariants mean dimension and entropy, see [18, 19]. If

s \in Γ

has infinite order, then

dim (R Γ (s - 1) | R Γ) = 1

and

dim (R Γ / R Γ (s - 1)) = 0

[18, Example 6.3].…”

Section: Bivariant Sylvester Module Rank Functionsmentioning

confidence: 99%

Bivariant and extended Sylvester rank functions

2020

J. London Math. Soc.

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For a unital ring R, a Sylvester rank function is a numerical invariant which can be described in three equivalent ways: on finitely presented left R‐modules, or on rectangular matrices over R, or on maps between finitely generated projective left R‐modules. We extend each Sylvester rank function to all pairs of left R‐modules scriptM1⊆scriptM2, and to all maps between left R‐modules satisfying suitable properties including continuity and additivity. As an application, we show that for any epimorphism R→S of unital rings, the pull‐back map from the set of Sylvester rank functions of S to that of R is injective. We also give a new proof of Schofield's result describing the image of this map when S is the universal localization of R inverting a set of maps between finitely generated projective left R‐modules.

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“…Define [18, Definition 3.1]

dim false(M_{1} false| M_{2} false) : = \underset{A \in F ({scriptM}_{1})}{trueprefixsup} \underset{B \in F ({scriptM}_{2})}{trueprefixinf} \underset{F \in F (Γ)}{trueprefixinf} \lim_{j \to ω} \frac{L (M false(scriptA, scriptB, F, σ_{j} false))}{false| X_{j} false|} .

Then

dim (\cdot | \cdot)

is a bivariant Sylvester module rank function for

R Γ

s \in Γ

has infinite order, then

dim (R Γ (s - 1) | R Γ) = 1

and

dim (R Γ / R Γ (s - 1)) = 0

[18, Example 6.3].…”

Section: Bivariant Sylvester Module Rank Functionsmentioning

confidence: 99%

Bivariant and extended Sylvester rank functions

2020

J. London Math. Soc.

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“…See also [49] and [58] for Z-actions and amenable group actions, respectively, on general locally compact Abelian groups. Finally, let us mention that an instance of the Bridge Theorem for actions of sofic groups on compact metrizable Abelian groups has been proved by Liang [39].…”

Section: Introductionmentioning

confidence: 97%

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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“…Ultrafilter have already been used several times in the study of measure entropy for actions of nonamenable groups (see [16,39,41,42,33,34]), and our paper is another entry into this tradition.…”

Section: Introductionmentioning

confidence: 98%

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

Hayes¹

2018

Preprint

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We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak * -limit of measures supported on topological microstates is equal to the smallest subgroup which absorbs all topological microstates. We are able to provide a version for the case when the centralizer is not ergodic. We give many applications, including show that for residually finite groups completely positive (lower) topological entropy (in the presence) is equivalent to completely positive (lower) measure entropy in the presence.

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Entropy on modules over the group ring of a sofic group

Abstract: We partially generalize Peters' formula [14] on modules over the group ring FΓ for a given finite field F and a sofic group Γ. It is also discussed that how the values of entropy are related to the zero divisor conjecture.

Cited by 4 publications

References 16 publications

Bivariant and extended Sylvester rank functions

Bivariant and extended Sylvester rank functions

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

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

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