On equations in free groups.

Steinberg, Arthur

doi:10.1307/mmj/1029000593

Cited by 11 publications

(3 citation statements)

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“…By way of contrast to our results we note that the theorem of Lewin and Steinberg (see [5] and [13]) applies to show that w = x k1 1 x k2 2 · · · x kn n has no nontrivial roots if n ≥ 3 and k 1 , . .…”

Section: Introductioncontrasting

confidence: 97%

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

McCool

2000

Int. J. Algebra Comput.

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Section: Introductioncontrasting

confidence: 97%

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

McCool

2000

Int. J. Algebra Comput.

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“…Thus, the equation has the following form in the new unknowns , ξ, η : (41) x ξ y η x ξ η −1 y −1 = c 2d P v B −1 P −1 v c −2d B.…”

Section: Daciberg L Gonc ¸Alveselena Kudryavtsevaheiner Zieschangmentioning

confidence: 99%

“…Equations in free groups have been extensively studied for many years: see Culler [8], Hmelevskiȋ [18,19,20], Lyndon [28,29], Lyndon and Schupp [30, Sections 1.6 and 1.8], Makanin [32], Razborov [37], Steinberg [41] and Wicks [46]; see also Gonçalves 220 Daciberg L GonçalvesElena KudryavtsevaHeiner Zieschang and Zieschang [13], Grigorchuk and Kurchanov [14], Grigorchuk, Kurchanov and Zieschang [15], Ol'shanskiȋ [34], Osborne and Zieschang [36], and Zieschang [47,48].…”

Section: Introductionmentioning

confidence: 99%

Some quadratic equations in the free group of rank 2

Gonçalves¹,

Kudryavtseva²,

Zieschang³

2008

Geometry and Topology Monographs

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For a given quadratic equation with any number of unknowns in any free group F , with right-hand side an arbitrary element of F , an algorithm for solving the problem of the existence of a solution was given by Culler [8] using a surface method and generalizing a result of Wicks [46]. Based on different techniques, the problem has been studied by the authors [11,12] for parametric families of quadratic equations arising from continuous maps between closed surfaces, with certain conjugation factors as the parameters running through the group F . In particular, for a one-parameter family of quadratic equations in the free group F 2 of rank 2, corresponding to maps of absolute degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence of faithful solutions has been solved in terms of the value of the self-intersection index µ : F 2 → Z[F 2 ] on the conjugation parameter. The present paper investigates the existence of faithful, or non-faithful, solutions of similar families of quadratic equations corresponding to maps of absolute degree 0. The existence results are proved by constructing solutions. The non-existence results are based on studying two equations in Z[π] and in its quotient Q, respectively, which are derived from the original equation and are easier to work with, where π is the fundamental group of the target surface, and Q is the quotient of the abelian group Z[π \ {1}] by the system of relations g ∼ −g −1 , g ∈ π \ {1}. Unknown variables of the first and second derived equations belong to π , Z[π], Q, while the parameters of these equations are the projections of the conjugation parameter to π and Q, respectively. In terms of these projections, sufficient conditions for the existence, or non-existence, of solutions of the quadratic equations in F 2 are obtained.20E05, 20F99; 57M07, 55M20, 20F05

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