The Jacobian algebra associated to a triangulation of a closed surface S
with a collection of marked points M is (weakly) symmetric and tame. We show
that for these algebras the Auslander-Reiten translate acts 2-periodical 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 four
properties implies that there is a new family of algebras symmetric, tame and
with periodic module category.
As a consequence of the 2-periodical actions of the Auslander-Reiten
translate on objects, we have that the Auslander-Reiten quiver of the
generalized cluster category \cC_{(S,M)} consists only of stable tubes of
rank 1 or 2.Comment: In the previous version I showed that the Jacobian algebras of closed
surfaces, excluding the case of the sphere with 4 and 5 punctures, are
algebras of exponential growth and it was changed grammar errors. In this new
version I change the name of this note and I show that the Jacobian algebras
of the sphere with 5 punctures are algebras of exponential growth. This is
the final versio
