scholarly journals Torsion, torsion length and finitely presented groups

2018 ◽  
Vol 21 (5) ◽  
pp. 949-971
Author(s):  
Maurice Chiodo ◽  
Rishi Vyas
Keyword(s):  
Embedding Theorem ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
New Construction ◽  
Finitely Presented ◽  
Torsion Free

Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

scholarly journals Direct products and properly 3-realisable groups

2004 ◽  
Vol 70 (2) ◽  
pp. 199-205 ◽  
Author(s):  
Manuel Cárdenas ◽  
Francisco F. Lasheras ◽  
Ranja Roy
Keyword(s):  
Homotopy Type ◽  
Universal Cover ◽  
Hyperbolic Groups ◽  
Direct Products ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Manifold With Boundary ◽  
Proper Homotopy ◽  
Finitely Presented

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover K̃ has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

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.

scholarly journals Torsion-free word hyperbolic groups are noncommutatively slender

2016 ◽  
Vol 26 (07) ◽  
pp. 1467-1482 ◽  
Author(s):  
Samuel M. Corson
Keyword(s):  
Free Subgroup ◽  
Hyperbolic Groups ◽  
Topological Groups ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Word Hyperbolic Groups ◽  
Hawaiian Earring ◽  
Free Word ◽  
Finitely Presented ◽  
Torsion Free

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

scholarly journals Matricial Representations of Certain Finitely Presented Groups Generated by Order-2 Generators and Their Applications

2017 ◽  
Vol 14 (3) ◽  
Author(s):  
Ryan Golden ◽  
Ilwoo Cho
Keyword(s):  
Group Algebra ◽  
Free Probability ◽  
Group Presentation ◽  
Lucas Numbers ◽  
Group Relations ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Finitely Presented ◽  
Representation Group ◽  
Presentation Group

In this paper, we study matricial representations of certain finitely presented groups Γ2Nwith N-generators of order-2. As an application, we consider a group algebra A2 of Γ22; under our representations. Specifically, we characterize the inverses g-1of all group elements g in Γ22; in terms of matrices in the group algebra A2. From the study of this characterization, we realize there are close relations between the trace of the radial operator of A2; and the Lucas numbers appearing in the Lucas triangle. KEYWORDS: Matricial Representation; Group Presentation; Group Algebras; Lucas Numbers; Lucas Triangle; Finitely Presented Group;Group Relations; Free Probability

Computing with knot quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850074
Author(s):  
Graham Ellis ◽  
Cédric Fragnaud
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Knot Invariant ◽  
Number Formula ◽  
Finitely Presented

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

On groups and 3-manifolds with weak dimension ≤ 1

1974 ◽  
Vol 75 (1) ◽  
pp. 33-35
Author(s):  
David E. Galewski
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Connected Sum ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Weak Dimension ◽  
Finitely Presented

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

scholarly journals INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 335-344
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Normal Subgroup ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Free Semigroups ◽  
Finitely Presented ◽  
Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

Subgroups of finitely presented groups

1961 ◽  
Vol 262 (1311) ◽  
pp. 455-475 ◽  
Keyword(s):  
Finite Group ◽  
Genetic Characterization ◽  
Recursively Enumerable Set ◽  
Locally Finite ◽  
Finitely Presented Groups ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented

The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

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