Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

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 from $R(\Lambda,V)$ to $R$ parametrize the lifts of $V$ up to $R\Lambda$, where $R$ is any appropriate complete, local commutative Noetherian $k$-algebra. 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 a symmetric special biserial algebra of polynomial growth which coincides with an acyclic Brauer graph algebra. 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 $R(\Lambda,V)$