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.

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.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

1998 ◽  
Vol 08 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Susan Garner Garille ◽  
John Meier
Keyword(s):  
Local Structure ◽  
Universal Cover ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Group ◽  
Finite Presentation ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

scholarly journals The computability of group constructions II

1973 ◽  
Vol 8 (1) ◽  
pp. 27-60 ◽  
Author(s):  
R.W. Gatterdam
Keyword(s):  
Characteristic Function ◽  
Word Problem ◽  
Elementary Functions ◽  
Recursively Enumerable Set ◽  
Finitely Presented Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Computational Level

Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

scholarly journals Topological properties of full groups

2009 ◽  
Vol 30 (2) ◽  
pp. 525-545 ◽  
Author(s):  
JOHN KITTRELL ◽  
TODOR TSANKOV
Keyword(s):  
Hilbert Space ◽  
Equivalence Relation ◽  
Separable Hilbert Space ◽  
Equivalence Relations ◽  
Full Group ◽  
Topological Properties ◽  
Finitely Generated ◽  
Dense Subgroup ◽  
Separable Group ◽  
Finitely Generated Group

AbstractWe study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups Fm and Fn if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

CIRCUITS, THE GROUPS OF RICHARD THOMPSON, AND coNP-COMPLETENESS

2006 ◽  
Vol 16 (01) ◽  
pp. 35-90 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Simple Group ◽  
Word Problem ◽  
Partial Transformation ◽  
Transformation Groups ◽  
Combinational Circuits ◽  
Finitely Generated ◽  
Finitely Presented Group ◽  
Thompson Groups ◽  
Higman Group ◽  
Finitely Presented

We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson–Higman group G3,1.

scholarly journals Subgroups of finitely presented metabelian groups

1973 ◽  
Vol 16 (1) ◽  
pp. 98-110 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.

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 Taming the hydra: The word problem and extreme integer compression

2018 ◽  
Vol 28 (07) ◽  
pp. 1299-1381
Author(s):  
W. Dison ◽  
E. Einstein ◽  
T. R. Riley
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Complexity Measure ◽  
Dehn Function ◽  
Dehn Functions ◽  
Fast Growing ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Direct Attack ◽  
Finitely Presented

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.

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