Algebras of amenable representation type and (dimensional) expansion

Algebras of amenable representation type, introduced by Elek, are characterized by every module having a large submodule that is the direct sum of modules which are small with respect to some $\eps > 0$ such that the quotient is also small in that respect. Motivated by respective results for string algebras and wild Kronecker quivers, Elek conjectured that an algebra is of amenable type iff it is tame. I will recall the amenability of the 2-Kronecker quiver and discuss consequences for tame hereditary algebras. Further, we will see a counterexample to the amenability of wild Kronecker quivers via dimension expanders. Introducing a weaker notion and using the interpretation functors for finitely controlled wild algebras of Gregory-Prest, we show that non of these algebras is of amenable type, supporting the conjecture.