Conjugacy in relatively extra-large Artin groups

Let A be an Artin group with standard generators X={x 1, …,x n}, n≥1 and defining graph Γ A. A \emph{standard parabolic subgroup} of A is a subgroup generated by a subset of X. For elements u and v of A we say (as usual) that u is conjugate to v by an element h of A if h −1 uh=v holds in A. Similarly, if K and L are subsets of A then K is conjugate to L by an element h of A if h −1 Kh=L. In this work we consider the conjugacy of elements and standard parabolic subgroups of a certain type of Artin groups. Results in this direction occur in occur in papers by Duncan, Kazachkov, Remeslennikov, Fenn, Dale, Jun, Godelle, Gonzalez-Meneses, Wiest, Paris, Rolfsen, for example. Of particular interest are centralisers of elements, and of standard parabolic subgroups, normalisers of standard parabolic subgroups and commensurators of parabolic subgroups. In this work we consider similar problems in a new class of Artin groups, introduced in the paper "On relatively extralarge Artin groups and their relative asphericity", by Juhasz, where the word problem is solved, among other things. Also, intersections of parabolic subgroups and their conjugates are considered.