SURJECTIVITY AND EQUIDISTRIBUTION OF THE WORD xaybONPSL(2, q) ANDSL(2, q)

Bandman, Tatiana; Garion, Shelly

doi:10.1142/s0218196712500178

Cited by 9 publications

(31 citation statements)

References 15 publications

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“…In particular, this conjecture holds for Engel words and for words of the form (see subsection 6.1 and [8,9]). One can therefore attempt to use the trace map method described in subsection 2.3 to find more words satisfying the above conjecture.…”

Section: Conjecture 714 (Shalev 1 )mentioning

confidence: 82%

“…This allows one to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes. This result can be viewed, on the one hand, as a refinement (in the SL 2 -case) of equidistribution theorems of [69] and [73] on general words and general Chevalley groups G, and, on the other hand, as a generalization of equidistribution theorems for some particular words: [41] (commutator words on any finite simple G), [9] (Engel words on SL 2 ), [8] (words of the form = on SL 2 ). Acting in the spirit of [41], we deduce a criterion for : SL 2 × SL 2 → SL 2 to be almost measure-preserving.…”

Section: Criteria For Equidistributionmentioning

confidence: 89%

“…By further recent results of Guralnick and Malle [53] and of Liebeck, O'Brien, Shalev and Tiep [81], some words of the form are known to be surjective on all finite simple groups. The particular case of surjectivity of these words on SL(2 ) was studied in [8] (see subsection 6.1).…”

Section: ∈ G} As the Set Of Values Of In Gmentioning

confidence: 99%

“…It was proved in [41] that the word = 2 2 ∈ F 2 , the commutator word = [ ] ∈ F 2 , as well as the words = [ 1 ] ∈ F , -fold commutators in any arrangement of brackets, are almost equidistributed on the family of finite simple non-abelian groups. The case G = SL(2 ) was studied in some more detail in [8,9,14], see Section 6.…”

Section: ∈ G} As the Set Of Values Of In Gmentioning

confidence: 99%

“…In [8] there is a precise description of the positive integers and prime powers for which the word map ( ) = is surjective on the group PSL(2 ) (and SL (2 )). The proof is based on the investigation of the trace map of positive words.…”

Section: If ≤ 4 Then Is Surjective For All Groups Psl(2 )mentioning

confidence: 99%

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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

2013

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We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

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Section: Conjecture 714 (Shalev 1 )mentioning

confidence: 82%

Section: Criteria For Equidistributionmentioning

confidence: 89%

Section: ∈ G} As the Set Of Values Of In Gmentioning

confidence: 99%

Section: ∈ G} As the Set Of Values Of In Gmentioning

confidence: 99%

Section: If ≤ 4 Then Is Surjective For All Groups Psl(2 )mentioning

confidence: 99%

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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

2013

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Beauville Surfaces and Probabilistic Group Theory

Garion

2015

Springer Proceedings in Mathematics &Amp; Statistics

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A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called "Beauville structure". They conjectured that all finite simple groups, except A5, admit such a structure. This conjecture has recently been proved by Guralnick-Malle and Fairbairn-Magaard-Parker.In this survey we demonstrate another approach towards the proof of this conjecture, based on probabilistic group-theoretical methods, by describing the following three works. The first is the work of Garion, Larsen and Lubotzky, showing that the above conjecture holds for almost all finite simple groups of Lie type. The second is the work of Garion and Penegini on Beauville structures of alternating groups, based on results of Liebeck and Shalev, and the third is the case of the group PSL2(p e ), in which we give bounds on the probability of generating a Beauville structure. We also discuss other related problems regarding finite simple quotients of hyperbolic triangle groups and present some open questions and conjectures.

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Some Results and Problems in the Theory of Word Maps

Shalev

2013

Bolyai Society Mathematical Studies

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SURJECTIVITY AND EQUIDISTRIBUTION OF THE WORD x^ay^bONPSL(2, q) ANDSL(2, q)

Cited by 9 publications

References 15 publications

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

Beauville Surfaces and Probabilistic Group Theory

Some Results and Problems in the Theory of Word Maps

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