Cocompact lattice actions on flag boundaries of Euclidean buildings are topologically stable: every sufficiently small perturbation is semi-conjugate to the original action.
Marked length spectrum rigidity for relatively hyperbolic groups
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abstract
We consider a coarse version of the marked length spectrum rigidity: given a group with two left invariant metrics, if the marked length spectrum (the translation length function) under the two metrics are the same, then the two metrics are uniformly close. We prove the rigidity theorem for relatively hyperbolic groups. This generalizes a result of Fujiwara.
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Stability for boundary actions of cocompact lattices in Euclidean buildings
Cocompact lattice actions on flag boundaries of Euclidean buildings are topologically stable: every sufficiently small perturbation is semi-conjugate to the original action.