Jacobian algebras with periodic module category and exponential growth

Recently it was proven by Geiss, Labardini-Fragoso and Sh¨oer in [1] that every Jacobian algebra associated to a triangulation of a closed surface S with a collection of marked points M is tame and Ladkani proved in [2] these algebras are (weakly) symmetric. In this work we show that for these algebras the Auslander-Reiten translation acts 2-periodically on objects. Moreover, we show that excluding only the case of a sphere with 4 (or less) punctures, these algebras are of exponential growth. These results imply that the existing characterization of symmetric tame algebras whose non-projective indecomposable modules are Ω-periodic, has at least a missing class (see [3, Theorem 6.2] or [4]). As a consequence of the 2-periodical actions of the Auslander-Reiten translation on objects, we have that the Auslander-Reiten quiver of the generalized cluster category C(S,M) consists only of stable tubes of rank 1 or 2.