Return Groups of Eventually Dendric Shifts

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.