Geometry and arithmetic of verbal dynamical systems on simple groups

2008/09/02 by Tatiana Bandman, Fritz Grunewald, Bandman, Tatiana +6
Mathematics · #20D10 #20G30 #37P05 #37P15 #37P35 #37P55 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #math.AG #math.DS #msc:20D10 #msc:20G30 #msc:37P05 #msc:37P15 #msc:37P35 #msc:37P55

paper · pdf · doi:10.48550/arxiv.0809.0369

35 pages; substantial revision; title changed; appendix by Nathan Jones added

openalex publication_date 2008/09/02 · arxiv created 2009/09/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.

Citations

• The Elementary Theory of the Frobenius Automorphisms

Related

• Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
• On Word Equations Originated from Discrete Dynamical Systems Related to\n Antisymmetric Cubic Maps With Some Applications
• Word and Conjugacy Problems in Groups G_k+1^k
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• Word Maps in Finite Simple Groups