Groupes de retour de shifts ultimement dendriques

Since 2015, dendric shifts (a generalisation of Sturmian words) have been widely studied. One of the results concerning these shift spaces is the return theorem. It describes the groups generated by the return words of a dendric shift. The proof uses the fundamental group of the Rauzy graph of the shift space. Later, eventually dendric shifts were introduced. They are of utmost importance because, unlike dendric shifts, they are stable under conjugacy. This key feature makes eventual dendricity a dynamical property. It seems natural to investigate if results similar to the return theorem hold in the eventually dendric case. The aim of this presentation is to introduce return groups and dendricity. It will also contain new results concerning the return theorem in the case of eventually dendric shifts and showcase the tools used to prove it

Eventually dendric shifts

International audienc

Groupes fondamentaux et dendricité

The aim of the poster is to showcase the interplay between group theory, algebraic topology and combinatorics on words. A result that allows to display this is the return theorem by Berté et al. in 2015. The poster will contain an introduction to fundamental groups of graphs, dendric words as well as a new result concerning return groups of eventually dendric shifts. More precisely, dendric words are words whose factors cannot be arbitrarily extended and their return groups are the groups of loops over their Rauzy graphs. These contain the structure of the entire word and so the specific structure induced by dendric words allows us to compute these groups explicitely