# Infinitely many Lagrangian fillings

### Abstract

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 which induce faithful actions of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ and the mapping class group $M_{0,4}$ 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.

## Authors

Roger Casals

Department of Mathematics, University of California Davis, Davis, CA

Honghao Gao

Department of Mathematics, Michigan State University, East Lansing, MI