Local $C^0$-semi-rigidity of meandering-hyperbolic actions

Abstract

Meandering hyperbolicity, introduced by Kapovich, Kim, and Lee, extends classical hyperbolicity beyond the setting of word-hyperbolic groups. In this paper, we prove that every meandering-hyperbolic action is locally semi-rigid in the $C^0$ topology. This extends the previously established stability theory in the Lipschitz $C^0$ topology to the full $C^0$ topology. Consequently, we recover the $C^0$-local semi-rigidity of both boundary actions of word-hyperbolic groups and cocompact lattices in semisimple Lie groups from a single dynamical principle.