The structure of one-relator relative presentations and their centres

Klyachko, Anton A.

doi:10.1515/jgt.2009.024

Cited by 9 publications

(12 citation statements)

References 22 publications

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“…Our arguments utilize diagrams over free products of groups and are based on a car‐crash lemma of , see also , that has had quite a few applications in group theory, see .…”

Section: Introductionmentioning

confidence: 99%

Quasiperiodic and mixed commutator factorizations in free products of groups

Ivanov

Klyachko

2018

Bull. London Math. Soc.

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It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation [x1, y1] . . . [x k , y k ] = z n , where n 2k, in the free product F of groups without nontrivial elements of order n implies that z is conjugate to an element of a free factor of F . If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct. m j=1 n j . Culler [8] discovered that, in the free group F (a, b) with free generators a, b, the cube [a, b] 3 of the commutator [a, b] := a −1 b −1 ab is a product of two commutators, [a, b] 3 = [a −1 ba, a −2 bab −1 ][bab −1 , b 2 ].Moreover, [a, b] n is a product of k commutators whenever n 2k − 1, see [8].

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“…Our arguments utilize diagrams over free products of groups and are based on a car‐crash lemma of , see also , that has had quite a few applications in group theory, see .…”

Section: Introductionmentioning

confidence: 99%

Quasiperiodic and mixed commutator factorizations in free products of groups

Ivanov

Klyachko

2018

Bull. London Math. Soc.

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“…The following lemma is an easy generalisation of Lemma 2.1 from [Le09]; a similar trick with the change of presentation was used in [Kl93] and later in many other works (see, e.g., [KP95], [CG95], [CG00], [CR01], [FeR96], [FeR98], [FoR05], [Kl05], [Kl06b], [Kl07], and [Kl09]). A geometric interpretation of this trick can be found in [FoR05].…”

Section: An Algebraic Lemmamentioning

confidence: 97%

“…-G embeds (naturally) into G [Kl93] (see also [FeR96]);* ) -G is torsion-free [FoR05]; -G is not simple if it does not coincide with G [Kl05]; -G almost always (with some known exceptions) contains a non-abelian free subgroup [Kl07]; -G is SQ-universal if G decomposes nontrivially into a free product [Kl06b]; -the centre of G is almost always (with some known exceptions) trivial [Kl09]. Some generalisations of these results to relative presentations with several additional generators can be found in [Kl09], [Kl07], [Kl06a], and [Kl06b].…”

Section: Introductionmentioning

confidence: 99%

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

Klyachko

Lurye

2012

Journal of Pure and Applied Algebra

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A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.Henceforth, the symbol |u| denotes the number of letters t ±1 in the word u.Relatively hyperbolic groups have many good properties. For example, they are SQ-universal (apart from some obvious exceptions) [AMO07], the word [Far98] and conjugacy [Bum04] problems are solvable in such groups (under some natural restrictions). The same is true for many other algorithmic problems. More details about relatively hyperbolic groups can be found in book [Os06].It turns out that the torsion-freeness condition in Le Thi Giang's theorem can be replaced by the absence of only order-two elements. Presently, the following theorem is the unique result about unimodular relative presentations in which torsion-freeness condition is weakened to the absence of small-order elements.

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“…Before investigating this question we explain why possible pinning may be interesting. The first point was suggested by Klyachko in [10], where he also introduced the effect of pinning.…”

Section: Generalized Pauli Constraintsmentioning

confidence: 99%

The quantum marginal problem

Schilling¹

2014

Mathematical Results in Quantum Mechanics

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The question of whether given density operators for subsystems of a multipartite quantum system are compatible to one common total density operator is known as the quantum marginal problem. We briefly review the solution of a subclass of such problems found just recently. In particular, this provides the solution of the 1-body N -representability problem. Its solution, the so-called generalized Pauli constraints, restrict the set of mathematically possible fermionic occupation numbers significantly, and strengthens Pauli's exclusion principle. Moreover, we review the study of a concrete physical model of interacting fermions confined to a harmonic trap. There, we found occupation numbers close, but not exactly on the boundary of the allowed region. This new effect of quasipinning is physically relevant since it corresponds to a simplified structure of the corresponding N -fermion quantum state.

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The structure of one-relator relative presentations and their centres

Abstract: Abstract. Suppose that G is a non-trivial torsion-free group and w is a word in the alphabet G U fx G1

Cited by 9 publications

References 22 publications

Quasiperiodic and mixed commutator factorizations in free products of groups

Quasiperiodic and mixed commutator factorizations in free products of groups

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

The quantum marginal problem

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