Let $A$ be a nite dimensional $k-$algebra and $R$ be a
locally bounded category such that $R rightarrow R/G = A$ is a Galois covering
de ned by the action of a torsion-free group of automorphisms
of $R$. Following [30], we provide criteria on the convex subcategories
of a strongly simply connected category R in order to be a cycle-
nite category and describe the module category of $A$. We provide
criteria for $A$ to be of polynomial growth
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