We show that two well-studied classes of tame algebras coincide: namely, the
class of symmetric special biserial algebras coincides with the class of Brauer
graph algebras. We then explore the connection between gentle algebras and
symmetric special biserial algebras by explicitly determining the trivial
extension of a gentle algebra by its minimal injective co-generator. This is a
symmetric special biserial algebra and hence a Brauer graph algebra of which we
explicitly give the Brauer graph. We further show that a Brauer graph algebra
gives rise, via admissible cuts, to many gentle algebras and that the trivial
extension of a gentle algebra obtained via an admissible cut is the original
Brauer graph algebra.
As a consequence we prove that the trivial extension of a Jacobian algebra of
an ideal triangulation of a Riemann surface with marked points in the boundary
is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs
of the triangulation.Comment: Minor changes, to appear in Journal of Algebr
