Résumé : I'll discuss the history of work on the question of how and whether rationality of growth depends on generators. That is, for a finitely generated group, one can study the growth function (the number of words spellable in up to n letters) and its growth series, the associated generating function. This growth series is itself a rational function when there is a recursive relationship among the values of the growth function. It is known that all virtually abelian groups and all hyperbolic groups have rational growth in any generators, by work of Benson and Cannon respectively. Work of Shapiro and of Stoll sheds some light on the situation in nilpotent groups, but shows it to be more complicated-- even for the second Heisenberg group H_5, some generators give rational growth but others do not. I'll describe work in progress with Shapiro giving geometric arguments to establish rationality in all generators for the classical Heisenberg group H_3.

Dernière modification : Thursday 21 November 2024

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