Conjugacy properties in classes of Artin groups and their extensions

Abstract

In this thesis we study conjugacy properties from a combinatorial, formal language and algorithmic perspective, in classes of Artin groups and their extensions. We study the nature of conjugacy languages for right-angled Artin groups and their extensions. We also solve the twisted conjugacy problem for right-angled Artin groups, with respect to length-preserving automorphisms, and consider various examples, using geometric and algebraic techniques, where extensions of right-angled Artin groups have solvable conjugacy problem. For dihedral Artin groups, we compute conjugacy geodesic representatives, and show that the conjugacy growth series is transcendental for dihedral Artin groups, with respect to some generating set. We also prove regularity of the conjugacy geodesic language, by studying the permutation conjugator length function and falsification by fellow traveller property. Finally, we solve the twisted conjugacy problem for all dihedral Artin groups, which leads to new examples of extensions of dihedral Artin groups with solvable conjugacy problem. Key words: Conjugacy growth, conjugacy languages, twisted conjugacy, right-angled Artin groups, graph products, dihedral Artin groups.