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

Tatiana Bandman, Shelly Garion, Boris Kunyavskiǐ

Research output: Contribution to journalReview articlepeer-review

10 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)175-211
Number of pages37
JournalCentral European Journal of Mathematics
Volume12
Issue number2
DOIs
StatePublished - 2014

Bibliographical note

Funding Information:
Bandman and Kunyavski˘ı were supported in part by the Minerva Foundation through the Emmy Noether Research Institute for Mathematics. Kunyavski˘ı was supported in part by the Israel Science Foundation, grant 1207/12; a part of this work was done when he participated in the trimester program Arithmetic and Geometry in the Hausdorff Research Institute for Mathematics (Bonn). Garion was supported by the SFB 878 Groups, Geometry and Actions. Support of these institutions is gratefully appreciated.

Keywords

  • Arithmetic dynamics
  • Finite fields
  • Finite simple groups
  • Lang-Weil estimate
  • Matrix groups
  • Periodic points
  • Special linear group
  • Trace map
  • Word map

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