In this talk, we present a conjugacy invariant on multidimensional subshifts, the projective fundamental group, highlighting the simlilarities with classical objects in mathematics. We then present a construction showing that a natural class of tilings of the grid, subshifts of finite type, realizes a large class of groups as their projective fundamental groups.

In this talk, we recall and motivate the definition of the fundamental group of a topological space. We show how this can be understood in the context of graphs, and present the notion of universal cover. We then define a natural class of subshifts, the Hom shifts, using graphs, and show how the fundamental group of the graph relates to the fundamental group of the Hom shift it generates.

This talk will try to highlight various links between some usual problems in symbolic dynamics, and the general study of a conjugacy invariant called the "topological full group". We do not give any new result, but illustrate with specific examples from symbolic dynamics some of the known properties of this group, and present some of the general methods used to prove these classical results.

A classical result from the theory of formal languages, the Myhill-Nerode theorem, gives a necessary and sufficient condition in terms of congruence classes for a language to be regular. In this talk, we try to adapt this result to the case of subshifts, in which we consider potentially multidimensional infinite configurations rather than finite words. In particular, we study the behavior of extender entropy, a property introduced by R.Pavlov and T.French which is analogous to congruence classes in formal languages, and obtain some computability characterizations on the possible extender entropies of various classes of subshifts.