Degrees of unsolvability of the conjugacy problem for finitely-presented groups

1968 ◽  
Vol 7 (6) ◽  
pp. 357-387 ◽  
Author(s):  
L. A. Bokut'
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

Degrees of unsolvability of the conjugacy problem for finitely-presented groups

1968 ◽  
Vol 7 (5) ◽  
pp. 284-329 ◽  
Author(s):  
L. A. Bokut'
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

scholarly journals SUBGROUPS OF FINITELY PRESENTED GROUPS WITH SOLVABLE CONJUGACY PROBLEM

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1075-1084 ◽  
Author(s):  
ALEXANDER YU. OLSHANSKII ◽  
MARK V. SAPIR
Keyword(s):  
Conjugacy Problem ◽  
Countable Group ◽  
Finitely Presented Groups ◽  
Power Problem ◽  
Finitely Presented

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

2009 ◽  
Vol 02 (04) ◽  
pp. 611-635 ◽  
Author(s):  
K. Kalorkoti
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Computation Step ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Solvable Word ◽  
Extended Notion ◽  
Ordered Pairs

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

CONJUGACY OF FINITE SUBSETS IN HYPERBOLIC GROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 725-756 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JAMES HOWIE
Keyword(s):  
Linear Function ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Time Algorithm ◽  
Hyperbolic Groups ◽  
Finitely Presented Groups ◽  
Curved Spaces ◽  
Negatively Curved ◽  
Finitely Presented ◽  
Torsion Free

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x-1 Ax = B for some x ∈ Γ with ǁxǁ less than a linear function of max {ǁγǁ : γ ∈ A ∪ B}. (The coefficients of this linear function depend only on k and δ.) These results have implications for group-based cryptography and the geometry of homotopies in negatively curved spaces. In an appendix, we give examples of finitely presented groups in which the conjugacy problem for elements is soluble but the conjugacy problem for finite lists is not.

scholarly journals Constructing matrix representations of finitely presented groups

1991 ◽  
Vol 12 (4-5) ◽  
pp. 427-438 ◽  
Author(s):  
S.A. Linton
Keyword(s):  
Matrix Representations ◽  
Finitely Presented Groups ◽  
Finitely Presented

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

scholarly journals Embedding Free Burnside Groups in Finitely Presented Groups

2005 ◽  
Vol 111 (1) ◽  
pp. 87-105 ◽  
Author(s):  
S. V. Ivanov
Keyword(s):  
Finitely Presented Groups ◽  
Burnside Groups ◽  
Finitely Presented
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