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

Gnutov, F.; Gordeev, Nikolai

doi:10.1007/s00013-020-01442-7

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“…Proof. The group G = SU 2 (C) has a finite subgroup H ≈ SL 2 (5). From the definition of BGGKPP-words, it follows that Im w |H ≮ Z(H) (here, w j|H is the restriction of the word map w j : G n → G to a quasisimple finite subgroup H).…”

Section: Examplementioning

confidence: 99%

“…. ]], the corresponding word maps w are surjective on compact Lie groups (there are a number of other examples on simple algebraic groups; see [2,5]). In the present paper, we give an example of a sequence {w j } j∈N of the free group F n such that w j ∈ F j n \ F j+1 n , and each word map w j : SU 2 (C) n → SU 2 (C) is surjective (Theorem 2, Corollary 7.1).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%