An algebra has a solvable word problem if and only if it is embeddable in a finitely generated simple algebra

1978 ◽  
Vol 8 (1) ◽  
pp. 197-204 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Word Problem ◽  
Simple Algebra ◽  
Finitely Generated ◽  
Solvable Word Problem ◽  
Solvable Word

On Weighted Geodesics in Groups

1987 ◽  
Vol 30 (1) ◽  
pp. 86-91
Author(s):  
Seymour Lipschutz
Keyword(s):  
Weight Function ◽  
Free Product ◽  
Word Problem ◽  
Minimum Weight ◽  
Minimum Length ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Geodesic Problem ◽  
Solvable Word

AbstractA word W in a group G is a geodesic (weighted geodesic) if W has minimum length (minimum weight with respect to a generator weight function α) among all words equal to W. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group G with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group G with a solvable weighted geodesic problem with respect to one weight function α1, but unsolvable with respect to a second weight function α2. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.

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).

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

HOMOLOGICAL DECISION PROBLEMS FOR FINITELY GENERATED GROUPS WITH SOLVABLE WORD PROBLEM

2002 ◽  
Vol 12 (01n02) ◽  
pp. 213-221 ◽  
Author(s):  
W. A. BOGLEY ◽  
J. HARLANDER
Keyword(s):  
Word Problem ◽  
Decision Problems ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Solvable Word

We show that for finitely generated groups P with solvable word problem, there is no algorithm to determine whether H1(P) is trivial, nor whether H2(P) is trivial.

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 Computability of Følner sets

2017 ◽  
Vol 27 (07) ◽  
pp. 819-830 ◽  
Author(s):  
Matteo Cavaleri
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Groups ◽  
Distortion Functions ◽  
Solvable Word Problem ◽  
Følner Sets ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Solvable Word

We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions — with subrecursive distortion functions — of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets. Moreover, we obtain some known and some new upper bounds for the Følner function for these particular extensions.

scholarly journals QUASI-ISOMETRY AND FINITE PRESENTATIONS OF LEFT CANCELLATIVE MONOIDS

2013 ◽  
Vol 23 (05) ◽  
pp. 1099-1114 ◽  
Author(s):  
ROBERT D. GRAY ◽  
MARK KAMBITES
Keyword(s):  
Word Problem ◽  
Main Tool ◽  
Geometric Characterization ◽  
Finitely Generated ◽  
Solvable Word Problem ◽  
Finite Presentability ◽  
Finite Presentations ◽  
Solvable Word ◽  
Cancellative Monoids

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

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