scholarly journals A generalization of the simulation theorem for semidirect products

2018 ◽  
Vol 39 (12) ◽  
pp. 3185-3206
Author(s):  
SEBASTIÁN BARBIERI ◽  
MATHIEU SABLIK
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
Word Problem ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Subshift Of Finite Type ◽  
Semidirect Products ◽  
Finitely Generated Group

We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y\subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\unicode[STIX]{x1D70E})$ such that the $H$-subaction of $(X,\unicode[STIX]{x1D70E})$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.

scholarly journals Operator algebras with hyperarithmetic theory

2020 ◽  
Author(s):  
Isaac Goldbring ◽  
Bradd Hart
Keyword(s):  
Word Problem ◽  
Operator Algebras ◽  
Cuntz Algebra ◽  
Finitely Generated ◽  
C Algebra ◽  
Finitely Generated Group ◽  
C Algebras ◽  
Existentially Closed ◽  
Solvable Word Problem ◽  
Solvable Word

Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

scholarly journals Cube Complexes, Subgroups of Mapping Class Groups and Nilpotent Genus

2017 ◽  
Author(s):  
Martin R. Bridson
Keyword(s):  
Finite Type ◽  
Mapping Class Groups ◽  
The Other ◽  
Mapping Class ◽  
Class Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Groups Of Automorphisms ◽  
Cube Complexes ◽  
Finitely Presented

Based on a lecture at PCMI this chapter is structured around two sets of results, one concerning groups of automorphisms of surfaces and the other concerning the nilpotent genus of groups. The first set of results exemplifies the theme that even the nicest of groups can harbour a diverse array of complicated finitely presented subgroups: we shall see that the finitely presented subgroups of the mapping class groups of surfaces of finite type can be much wilder than had been previously recognised. The second set of results fits into the quest to understand which properties of a finitely generated group can be detected by examining the group’s finite and nilpotent quotients and which cannot.

scholarly journals FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION

2015 ◽  
Vol 100 (1) ◽  
pp. 108-123 ◽  
Author(s):  
ANDREW PENLAND ◽  
ZORAN ŠUNIĆ
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Binary Tree ◽  
Rooted Tree ◽  
Finitely Generated ◽  
Subshift Of Finite Type ◽  
Regular Tree ◽  
Pattern Size ◽  
Tree Automorphisms ◽  
Pattern Group

We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.

scholarly journals On the computability of rotation sets and their entropies

2018 ◽  
Vol 40 (2) ◽  
pp. 367-401 ◽  
Author(s):  
MICHAEL A. BURR ◽  
MARTIN SCHMOLL ◽  
CHRISTIAN WOLF
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
Topological Entropy ◽  
Metric Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Subshift Of Finite Type ◽  
Continuous Potential ◽  
Rotation Set ◽  
Entropy Functions

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

Context-sensitive languages and G-automata

2017 ◽  
Vol 27 (02) ◽  
pp. 237-249
Author(s):  
Rachel Bishop-Ross ◽  
Jon M. Corson ◽  
James Lance Ross
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Closure Properties ◽  
Finitely Generated Group ◽  
Context Sensitive ◽  
The Family

For a given finitely generated group [Formula: see text], the type of languages that are accepted by [Formula: see text]-automata is determined by the word problem of [Formula: see text] for most of the classical types of languages. We observe that the only exceptions are the families of context-sensitive and recursive languages. Thus, in general, to ensure that the language accepted by a [Formula: see text]-automaton is in the same classical family of languages as the word problem of [Formula: see text], some restriction must be imposed on the [Formula: see text]-automaton. We show that restricting to [Formula: see text]-automata without [Formula: see text]-transitions is sufficient for this purpose. We then define the pullback of two [Formula: see text]-automata and use this construction to study the closure properties of the family of languages accepted by [Formula: see text]-automata without [Formula: see text]-transitions. As a further consequence, when [Formula: see text] is the product of two groups, we give a characterization of the family of languages accepted by [Formula: see text]-automata in terms of the families of languages accepted by [Formula: see text]- and [Formula: see text]-automata. We also give a construction of a grammar for the language accepted by an arbitrary [Formula: see text]-automaton and show how to get a context-sensitive grammar when [Formula: see text] is finitely generated with a context-sensitive word problem and the [Formula: see text]-automaton is without [Formula: see text]-transitions.

scholarly journals ON GENERATORS AND PRESENTATIONS OF SEMIDIRECT PRODUCTS IN INVERSE SEMIGROUPS

2009 ◽  
Vol 79 (3) ◽  
pp. 353-365 ◽  
Author(s):  
E. R. DOMBI ◽  
N. RUŠKUC
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Semidirect Products ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractIn this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

Omitting quantifier-free types in generic structures

1972 ◽  
Vol 37 (3) ◽  
pp. 512-520 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Prove Theorem ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Closed Groups ◽  
Central Result ◽  
Generic Structures ◽  
Solvable Word

The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].In [10] Neumann proved that if a finitely generated group H is recursively absolutely presentable then H is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group H is recursively absolutely presentable if and only if H can be recursively presented with solvable word-problem. Therefore, if a finitely generated group H can be recursively presented with solvable word-problem then H is embeddable in all nontrivial algebraically-closed groups.The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group H is embeddable in all non-trivial algebraically-closed groups then H can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable K-generic structures.We gratefully acknowledge many helpful conversations with Harry Simmons.

scholarly journals Group-theoretical graph categories

2021 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Symmetric Group ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Special Case

AbstractThe semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum–Weber, 2015) and skew categories of partitions (more general; due to Maaßen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph.

scholarly journals An algebraic characterization of groups with soluble word problem

1974 ◽  
Vol 18 (1) ◽  
pp. 41-53 ◽  
Author(s):  
William W. Boone ◽  
Graham Higman
Keyword(s):  
Word Problem ◽  
Focal Point ◽  
Algebraic Characterization ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Necessary And Sufficient ◽  
Finitely Presented

The following theorem is the focal point of the present paper. It stipulates an algebraic condition equivalent, in any finitely generated group, to the solubility of the word problem.THEOREM I. A necessary and sufficient condition that a finitely generated group G have a soluble word problem is that there exist a simple group H, and a finitely presented group K, such that G is a subgroup of H, and H is a subgroup of K.

scholarly journals Calibrating word problems of groups via the complexity of equivalence relations

2016 ◽  
Vol 28 (3) ◽  
pp. 457-471 ◽  
Author(s):  
ANDRÉ NIES ◽  
ANDREA SORBI
Keyword(s):  
Word Problem ◽  
Equivalence Relation ◽  
Word Problems ◽  
Truth Table ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented ◽  
Computably Enumerable

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

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