Surjectivity of certain word maps on $${{\mathrm{PSL}}}(2,\mathbb {C})$$ PSL ( 2 , C ) and $${{\mathrm{SL}}}(2,\mathbb {C})$$ SL ( 2 , C )

Bandman, Tatiana; Zarhin, Yuri G.

doi:10.1007/s40879-016-0101-9

Cited by 14 publications

(1 citation statement)

References 21 publications

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“…This conjecture is still widely open, there being only several partial results, see [6,10,45,46,47,65]. It is a special case of the conjecture in [57,Question 2].…”

Section: Evaluations Of Wordsmentioning

confidence: 99%

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Kanel-Belov¹,

Malev

Rowen

et al. 2020

SIGMA

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Let p be a polynomial in several non-commuting variables with coefficients in a field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set M n (K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl n (K) of matrices of trace 0, or all of M n (K). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for n = 2 in Section 2, some decisive results for n = 3 in Section 3, and partial information for n ≥ 3 in Section 4, also for non-multilinear polynomials. In addition we consider the case of K not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

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“…This conjecture is still widely open, there being only several partial results, see [6,10,45,46,47,65]. It is a special case of the conjecture in [57,Question 2].…”

Section: Evaluations Of Wordsmentioning

confidence: 99%

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Kanel-Belov¹,

Malev

Rowen

et al. 2020

SIGMA

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Recursive sequences of surjective word maps for the algebraic groups $$\mathrm {PGL}_2$$ and $${{\text {SL}}}_2$$

Gnutov

Gordeev

2020

Arch. Math.

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On Sequences of Word Maps of Compact Topological Groups

Gordeev

2022

J Math Sci

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In 2013, A. Thom proved that for any standard unitary group SU(C) (the compact form) and for any real number > 0, there is a nontrivial word w(x, y) in two variables such that the image of the word map w : SU n (C) 2 → SU n (C) is contained in -neighborhood of the identity of the group SU n (C). Actually, in Thom's paper there is a construction of a sequence {w j } j∈N , where w j ∈ F 2 , that converges uniformly on a compact group to the identity. In the present paper, a method for constructing of such sequences is proposed. Also, with the help of results obtained by T. Bandman et.al (2006), a sequence of surjective word maps w j : SU 2 (C) n → SU 2 (C) is constructed, where each word w j is contained in the corresponding member F j n of the derived series of the free group F n . Some comments and remarks related to such results and general properties of word maps of compact groups are given. Bibliography: 21 titles.

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Surjectivity of certain word maps on $${{\mathrm{PSL}}}(2,\mathbb {C})$$ PSL ( 2 , C ) and $${{\mathrm{SL}}}(2,\mathbb {C})$$ SL ( 2 , C )

Cited by 14 publications

References 21 publications

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Recursive sequences of surjective word maps for the algebraic groups $$\mathrm {PGL}_2$$ and $${{\text {SL}}}_2$$

On Sequences of Word Maps of Compact Topological Groups

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