Super-decomposable pure-injective modules over algebras with strongly simply connected Galois coverings

Assume that k is a field of characteristic different from 2, R is a strongly simply connected locally bounded k-category, G is an admissible group of k-linear automorphisms of R   and A=R/GA=R/G. We show that if A is not of polynomial growth, then there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of M. Ziegler that there exists a super-decomposable pure-injective A-module, if the field k is countable. We also prove that if A is of non-polynomial growth and R does not contain a convex subcategory which is hypercritical or pg-critical, then there exists a factor algebra B of A such that B is a string algebra of non-polynomial growth. This fact is one of the key elements in the proof of our main results.