# The structure of quasi-transitive graphs avoiding a minor with applications to the Domino Conjecture

Ugo Giocanti (G-Scop, Univ. Grenoble)An infinite graph is quasi-transitive if the action of its automorphism group on its vertex set has finitely many orbits. Roughly speaking, this means that the graph has a lot of symmetries. Starting with the work of Maschke (1896), a lot of work have been done on the structure of planar Cayley graphs,and more generally of planar quasi-transitive graphs. On the opposite, only few research has been done about the more general class of minor-excluded quasi-transitive graphs.

In this talk, I will present a structure theorem for such graphs, which is reminiscent of the Robertson-Seymour Graph MinorStructure Theorem. The proof of our result is mainly based on a combination of the work of Thomassen (1992) together with an extensive study of Grohe (2016) on the properties of separations of order 3 in finite graphs. Our proof involves some technical notions from structural graph theory and I will spend some time to present some of the key concepts involved and especially how they must be adapted to take into account the symmetries of the studied graph.

Eventually I will explain how such a result can be used to prove the so called domino problem conjecture for minor-excluded groups, extending previous results from Berger (1966) and Aubrun, Barbieri and Moutot (2019). I will also spend time to present other applications both at the group and at the graph level.

This is a joint work with Louis Esperet and Clément Legrand-Duchesne.