Tame hereditary path algebras and amenability

In this note we are concerned with the notion of amenable representation type as defined in a recent paper by G\'abor Elek. Roughly speaking, an algebra is of amenable type if for all $\varepsilon > 0$, every finite-dimensional module has a submodule which is a direct sum of modules which are small with respect to $\varepsilon$ such that the quotient is also small in that respect. We will show that the tame hereditary path algebras of quivers of extended Dynkin type over any field $k$ are of amenable type, thus extending a conjecture in the aforementioned paper to another class of tame algebras. In doing so, we avoid using already known results for string algebras. We also show that path algebras of wild acyclic quivers over finite fields are not amenable.