Algebraic and geometric aspects of two-dimensional Artin groups

Abstract

In this thesis we study the algebra and the geometry of two-dimensional Artin groups under various aspects. First, we solve the problem of acylindrical hyperbolicity, by proving that all the two-dimensional Artin groups that are not trivially non-acylindrically-hyperbolic are acylindrically hyperbolic. In particular, we prove that every non-spherical Artin group of dimension 2 has trivial centre. Then, we study the structure of parabolic subgroups of large-type Artin groups, and prove various results about their combinatorial structure. We notably show that any intersection of parabolic subgroups is again a parabolic subgroup. Finally, we study the isomorphisms between Artin groups of large-type, and we prove that the family of large-type free-of-infnity Artin groups is rigid. We also fully describe the automorphism groups of these Artin groups.