One-relater groups with center

Meskin, Stephen; Pietrowski, Alfred; Steinberg, Arthur

doi:10.1017/s1446788700015111

Cited by 11 publications

(16 citation statements)

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“…We remark that it is known that the group G = G a1,...,am is a two-generator group if and only if a i and a j are not coprime for some i = j [6]. Therefore for example, the isolated ordering of G 2,3,4 in Example 3.1 (ii) is an isolated ordering which is not derived from a Dehornoy-like ordering.…”

Section: M)mentioning

confidence: 97%

Construction of isolated left orderings via partially central cyclic amalgamation

Ito¹

2016

Tohoku Math. J. (2)

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Abstract. We give a new method to construct isolated left orderings of groups whose positive cones are finitely generated. Our construction uses an amalgamated free product of two groups having an isolated ordering. We construct a lot of new examples of isolated orderings, and give an example of isolated left orderings having various properties which previously known isolated orderings do not have. IntroductionThe set of all left orderings of G is denoted by LO(G). For g ∈ G, let U g be a subset of LO(G) defined byWe equip a topology on LO(G) so that {U g } g∈G is an open sub-basis of the topology. This topology is understood as follows. For a left ordering < G of G, G is decomposed as a disjoint union G = P (< G ) ⊔ {1} ⊔ P (< G ) −1 using the positive cone P (< G ). Conversely, a sub-semigroup P of G having this property is a positive cone of a left ordering of G: An ordering < P defined by g < P g ′ if g −1 g ′ ∈ P is a leftordering whose positive cone is P . Thus LO(G) is identified with a subset of the powerset 2 G−{1} . The topology of LO(G) defined as above coincides with the relative topology as the subspace of 2 G−{1} , equipped with the powerset topology. In this paper, we always consider countable groups, so we simply refer a countable group as a group unless otherwise specified. Then it is known that LO(G) is a compact, metrizable, and totally disconnected [10]. Moreover, LO(G) is either uncountable or finite [5]. Thus as a topological space, LO(G) is rather similar to the Cantor set: The main difference is that the space LO(G) might be non-perfect, that is, LO(G) might have isolated points. Indeed, if LO(G) has no isolated points and is not a finite set, then LO(G) is homeomorphic to the Cantor set. We call a left ordering which is an isolated point of LO(G) an isolated ordering.It is known that a left ordering < G whose positive cone is a finitely generated semigroup is isolated. In this paper we will concentrate our attention to study such an isolated ordering. We say a finite set of non-trivial elements of G, G = {g 1 , . . . , g r } defines an isolated left ordering < G of G if the positive cone of < G is generated by G as a semigroup. For an isolated left ordering < G of a group G, the 2010 Mathematics Subject Classification. Primary 20F60 , Secondary 06F15.

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Section: M)mentioning

confidence: 97%

Construction of isolated left orderings via partially central cyclic amalgamation

Ito¹

2016

Tohoku Math. J. (2)

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“…If G' is not a one-relator group then G is not a one-relator product and we are done. If G' is a one-relator group, then by Theorem 1 in [7] we have that (/?,, q f ) = 1 for every 1 <j<i<n.…”

Section: And B Finitely Generated Locally Indicable Groups With Non-tmentioning

confidence: 99%

“…Although it is not explicitly mentioned in [7], the problem Meskin, Pietrowski and Steinberg deal with is when the groups (1) are one-relator groups in the generating set {x,,x n+ ,}. In [7] a necessary condition and a sufficient condition are given on the ordered set of integers (p t , q t ,..., p n , q n ) for (1) to be a one-relator group but not a necessary and sufficient condition. Collins in [4] shows that any generating pair of G is Nielsen equivalent to a pair W,xj; +1 } with some extra conditions for r and s. Finally, McCool in [6] proves that if G is of the form (1) with p, = q t for every i -1,..., n + 1 then G is a one-relator group if H can be obtained from a suitable group (a, /?…”

Section: Introductionmentioning

confidence: 99%

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An algorithm for stem products and one-relator groups

Metaftsis¹

1999

Proceedings of the Edinburgh Mathematical Society

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“…, a pm−1 m−1 = a qm−1 m , a s m = a 1 m+1 . The conditions of part (b) of Theorem 1 of Meskin, Pietrowski and Steinberg [10] are satisfied, (with P 1 , . .…”

Section: Proof Of Theoremmentioning

confidence: 99%

FREE GROUP ROOTS OF a^kb^l AND [ a^k,b]

McCool

2000

Int. J. Algebra Comput.

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One-relater groups with center

Abstract: Communicated by M. F. Newman ABSTRACT. Many one-relator groups with center have been shown to be of the form -A necessary and a sufficient condition for the sequence (Pi,Qi,P 2 >Q2> '",Pt>Qt) a r e given in order for groups of the above form to be one-relator groups.

Cited by 11 publications

References 3 publications

Construction of isolated left orderings via partially central cyclic amalgamation

Construction of isolated left orderings via partially central cyclic amalgamation

An algorithm for stem products and one-relator groups

FREE GROUP ROOTS OF a^kb^l AND [ a^k,b]

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One-relater groups with center

Abstract: Communicated by M. F. Newman ABSTRACT. Many one-relator groups with center have been shown to be of the form -A necessary and a sufficient condition for the sequence (Pi,Qi,P 2 >Q2> '",Pt>Qt) a r e given in order for groups of the above form to be one-relator groups.

Cited by 11 publications

References 3 publications

Construction of isolated left orderings via partially central cyclic amalgamation

Construction of isolated left orderings via partially central cyclic amalgamation

An algorithm for stem products and one-relator groups

FREE GROUP ROOTS OF akbl AND [ ak,b]

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FREE GROUP ROOTS OF a^kb^l AND [ a^k,b]