THÈSE de DOCTORAT de l'UNIVERSITÉ PARIS 6 Litp /
Litp research reports
69 pages - Juin/June 1995 - French document.
PostScript : Ko /Kb
Titre / Title: COMBINATOIRE DES GROUPES À CROISSANCE POLYNOMIALE
Abstract : In this thesis we study from a combinatorial point of view Michael Gromov's theorem on groups of polynomial growth. We make a survey on his geometrical proof and present a construction which allows to visualize certain metric spaces he defines in it. Then we deal with the problem of giving a combinatorial proof. We show that there is a qualitative difference
between the case of linear growth and that of higher growth. We show that it comes out from a problem of periodicity of infinite words. This point of view allows us to show Gromov's result in the linear case with a combinatorial argument. To treat the case of higher growth, we introduce a property which we call rigidity. It allows us to transform an important part the algebraic problem in a problem of combinatorics on words. We use it to show in a completely combinatorial way that a group whose growth function is less or equal to a certain quadratic function cannot be periodic. In the final part of the thesis we extend our results to semigroups.
Publications internes Litp 1995 / Litp research reports 1995