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We prove that all maximal-tb positive Legendrian torus links (n, m) in the standard contact 3-sphere, except for (2, m), (3, 3), (3, 4) and (3, 5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL (2, Z) and the mapping class group M₀, ₄ into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.
Casals et al. (Sat,) studied this question.