Decision problems in classes of group presentations with uniformly solvable word problem

1981 ◽  
Vol 37 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Jody Lockhart
Keyword(s):  
Word Problem ◽  
Decision Problems ◽  
Group Presentations ◽  
Solvable Word Problem ◽  
Solvable Word

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.

Unsolvable Problems in Groups With Solvable Word Problem

1970 ◽  
Vol 22 (4) ◽  
pp. 836-838 ◽  
Author(s):  
James McCool
Keyword(s):  
Word Problem ◽  
Conjugacy Problem ◽  
The Other ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
Other Hand ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Effective Procedures ◽  
Solvable Word

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

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.

scholarly journals 2-Stratifold groups have solvable word problem

2017 ◽  
Vol 112 (3) ◽  
pp. 803-810 ◽  
Author(s):  
J. C. Gómez-Larrañaga ◽  
F. González-Acuña ◽  
Wolfgang Heil
Keyword(s):  
Word Problem ◽  
Solvable Word Problem ◽  
Solvable Word

Varieties of lie algebras with solvable word problem

1993 ◽  
Vol 21 (10) ◽  
pp. 3571-3609
Author(s):  
O. Kharlampovich ◽  
D. Gildenhuys
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Solvable Word Problem ◽  
Solvable Word

On recognising properties of groups which have solvable word problem

1970 ◽  
Vol 21 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Donald J. Collins
Keyword(s):  
Word Problem ◽  
Solvable Word Problem ◽  
Solvable Word
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