Wang's tiles were introduced in the 1960s and have been an inexhaustible source of undecidable problems ever since. They are unit square tiles with colored edges and fixed orientation, which can be placed together provided they share the same color on their common edge. Many decision problems involving Wang tiles follow the same global structure: given a finite set of Wang tiles, is there an algorithm to determine if they tile a particular shape or subset of the infinite grid? If we look for a tiling of the whole grid, this is the domino problem which is known to be undecidable for Z2 and many other groups. In this talk we focus on infinite snake tilings. Originally the infinite snake problem asks is there exists a tiling of a self-avoiding bi-infinite path on the grid Z2. In this talk I present how to expand the scope of domino snake problems to finitely generated groups to understand how the underlying structure affects computability. This is joint work with Nicolás Bitar.[-]

Right-angled Artin groups, aka partially commutative groups, naturally define an interpolation between free groups and abelian free groups. The mini-course is dedicated to the question: given two right-angled Artin groups, how can we know whether one is isomorphic to a subgroup of the other? Even though this is a basic algebraic question, it remains widely open in full generality. Our goal will be to show how the combinatorial geometry of quasi-median graphs hilights some aspects of this problem. [-]

I will present a general notion of automatic action, based on Büchi automata, and show how it unifies a large number of subclasses, in particular the automatic groups by Cannon, Thurston et al., the transducer groups by Aleshin, Grigorchuk, Sushchansky, Sidki et al., and substitutional subshifts. I will present some algorithms for these groups, and in particular show under an extra condition (boundedness) that their orbit relation is computable. This will have strong decidability consequences, such as that the order problem, aperiodicity, minimality, etc. for automatic transformations is decidable.[-]