Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups

We study the interplay between the algebraic and dynamical properties of groups that admit a general type action on a $\delta$-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful. We divide the class of such groups into two subclasses based on a dynamical criterion: groups whose induced action on the limit set is topologically free, and those whose action is not. We prove that satisfying the criterion is equivalent to a purely algebraic property of being mixed identity free, generalizing results of P. Fima, F. Le Ma\^{i}tre, S. Moon, and Y. Stalder from groups acting on trees to groups acting on arbitrary hyperbolic spaces. As one of the corollaries, we obtain that the reduced $C^*$-algebra of such groups is selfless, using the results of N. Ozawa. For the subclass of groups whose action on the limit set is not topologically free, we give the rigidity result for all $3$-transitive faithful actions and bound the transitivity degree by $3$, generalizing the result of A. Le Boudec and N. Matte Bon. As an application, we show that non-affine simple Kac-Moody groups over finite fields are MIF, answering the question of J. Belk, F. Fournier-Facio, J. Hyde, and M. Zaremsky.