Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems

We determine the maximum ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems. To obtain the upper bounds, we use the cubical flat torus theorem of Wise and Woodhouse. We then verify the sharpness of these upper bounds by constructing free abelian subgroups of the corresponding ranks. Our result shows that the maximal ranks of free abelian subgroups of ordinary cactus groups can be larger than conjectured by Genevois. As an application, we show that these groups are not quasi-isometric to any right-angled Artin group unless they are virtually free abelian.