Braid loops with infinite monodromy on the Legendrian contact DGA

We present the first examples of elements in the fundamental group of the space of Legendrian links in $(S^3,{\xi }_{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These new families include the first-known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer-theoretic proof that Legendrian $(n,m)$ torus links have infinitely many Lagrangian fillings if $n⩾3,m⩾6$ or $(n,m)=(4,4),(4,5)$. In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.