scholarly journals ON GROUPS AND COUNTER AUTOMATA

2008 ◽  
Vol 18 (08) ◽  
pp. 1345-1364 ◽  
Author(s):  
MURRAY ELDER ◽  
MARK KAMBITES ◽  
GRETCHEN OSTHEIMER
Keyword(s):  
Abelian Group ◽  
Word Problem ◽  
Word Problems ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Precise Sense

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller–Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognized by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata.

Automata with Counters that Recognize Word Problems of Free Products

2015 ◽  
Vol 26 (01) ◽  
pp. 79-98 ◽  
Author(s):  
Jon M. Corson ◽  
Lance L. Ross
Keyword(s):  
Free Product ◽  
Word Problem ◽  
Word Problems ◽  
Fundamental Theory ◽  
Finitely Generated ◽  
Free Products ◽  
Finitely Generated Groups ◽  
Bicyclic Monoid ◽  
The Family ◽  
Rees Quotient

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

The word problem for semigroups satisfying x3 = x

1978 ◽  
Vol 84 (1) ◽  
pp. 11-19 ◽  
Author(s):  
J. A. Gerhard
Keyword(s):  
Word Problem ◽  
Upper Bound ◽  
Finitely Generated ◽  
Finitely Generated Groups

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

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 A Descriptive View of Combinatorial Group Theory

2011 ◽  
Vol 17 (2) ◽  
pp. 252-264
Author(s):  
Simon Thomas
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Embedding Theorem ◽  
Combinatorial Group Theory ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Combinatorial Group

AbstractIn this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman–Neumann–Neumann Embedding Theorem.

On the Structure of Q2(G) for Finitely Generated Groups

1973 ◽  
Vol 25 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Symmetric Power ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

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.

Hierarchy for groups acting on hyperbolic ℤn-spaces

2021 ◽  
pp. 1-28
Author(s):  
Andrei-Paul Grecianu ◽  
Alexei Myasnikov ◽  
Denis Serbin
Keyword(s):  
Abelian Group ◽  
Systematic Study ◽  
Metric Spaces ◽  
Free Groups ◽  
Lexicographic Order ◽  
Finitely Generated ◽  
Princeton University ◽  
Finitely Generated Groups ◽  
The One ◽  
The Right

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text][Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

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.

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