  Let $k$ be an arbitrary field, $\Lambda$ be a $k$-algebra and $V$ be a
$\Lambda$-module. When it exists, the universal deformation ring $R(\Lambda,V)$
of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts
of $V$ up to $R\otimes_k \Lambda$, where $R$ is any complete, local commutative
Noetherian $k$-algebra with residue field $k$. Symmetric special biserial
algebras, which coincide with Brauer graph algebras, can be viewed as
generalizing the blocks of finite type $p$-modular group algebras. Bleher and
Wackwitz classified the universal deformation rings for all modules for
symmetric special biserial algebras with finite representation type. In this
paper, we begin to address the tame case. Specifically, let $\Lambda$ be any
1-domestic, symmetric special biserial algebra. By viewing $\Lambda$ as
generalized Brauer tree algebras and making use of a derived equivalence, we
classify the universal deformation rings for those $\Lambda$-modules $V$ with
stable endomorphism ring isomorphic to $k$. The latter is a natural condition,
since it guarantees the existence of the universal deformation ring
$R(\Lambda,V)$.
