Groups, Geometry, and Dynamics


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Volume 4, Issue 4, 2010, pp. 607–655
DOI: 10.4171/GGD/98

Published online: 2010-10-15

Geometry and arithmetic of verbal dynamical systems on simple groups

Tatiana Bandman[1], Fritz Grunewald[2], Boris Kunyavskii[3] and Nathan Jones[4]

(1) Bar-Ilan University, Ramat Gan, Israel
(2) Heinrich-Heine-Universität, Düsseldorf, Germany
(3) Bar-Ilan University, Ramat Gan, Israel
(4) University of Mississippi, USA

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.

Keywords: Solvable group, word map, trace map, dynamical system, periodic point, elliptic curve

Bandman Tatiana, Grunewald Fritz, Kunyavskii Boris, Jones Nathan: Geometry and arithmetic of verbal dynamical systems on simple groups. Groups Geom. Dyn. 4 (2010), 607-655. doi: 10.4171/GGD/98