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 language | English |
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Pages (from-to) | 175-211 |
Number of pages | 37 |
Journal | Central European Journal of Mathematics |
Volume | 12 |
Issue number | 2 |
DOIs | |
State | Published - 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.
Funding
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.
Funders | Funder number |
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Emmy Noether Research Institute for Mathematics | |
Hausdorff Research Institute for Mathematics | |
SFB 878 Groups, Geometry and Actions | |
Minerva Foundation | |
Israel Science Foundation | 1207/12 |
Keywords
- Arithmetic dynamics
- Finite fields
- Finite simple groups
- Lang-Weil estimate
- Matrix groups
- Periodic points
- Special linear group
- Trace map
- Word map