Parabolic subgroups of two-dimensional Artin groups and systolic-by-function complexes

We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large-type Artin groups to a broader family of two-dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a $(2,2)$-free two-dimensional Artin group is itself a parabolic subgroup. An Artin group is $(2,2)$-free if its defining graph does not have two consecutive edges labeled by $2$. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by considering systolic-by-function complexes, which generalize systolic complexes. Systolic-by-function complexes have a more flexible structure than systolic complexes since we allow the edges to have different lengths. At the same time, their geometry is rigid enough to satisfy an analogue of the Cartan-Hadamard theorem and other geometric properties similar to those of systolic complexes.