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

Tatiana Bandman, Shelly Garion, Boris Kunyavskiĭ (2014)

Open Mathematics

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.

Estimates for simple random walks on fundamental groups of surfaces

Laurent Bartholdi, Serge Cantat, Tullio Ceccherini-Silberstein, Pierre de la Harpe (1997)

Colloquium Mathematicae

Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.

Estimations de dimensions de Minkowski dans l’espace des groupes marqués

Luc Guyot (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cet article, on montre que l’espace des groupes marqués est un sous-espace fermé d’un ensemble de Cantor dont la dimension de Hausdorff est infinie. On prouve que la dimension de Minkowski de cet espace est infinie en exhibant des sous-ensembles de groupes marqués à petite simplification dont les dimensions de Minkowski sont arbitrairement grandes. On donne une estimation des dimensions de Minkowski de sous-espaces de groupes à un relateur. On démontre enfin que les dimensions de Minkowski...

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