PUB - Publications at Bielefeld University

Motivated by the introduction of the notion of amenable representation type by Elek [Ele17](https://doi.org/10.1007/s00208-017-1552-0), we study the conjecture that finite dimensional algebras are of tame representation type if and only if they are of amenable representation type. While string algebras were shown to be amenable in [Ele17](https://doi.org/10.1007/s00208-017-1552-0), in this thesis we prove that tame hereditary algebras, in particular path algebras of extended Dynkin quivers, are of amenable type.
Further results concern the amenability of tame concealed algebras as well as partial positive results for indecomposable modules of integral slope for tubular canonical algebras and the preprojective and postinjective component for path algebras of generalised Kronecker quivers.
To prove the other direction of the conjecture for algebraically closed fields, one may show that wild algebras are not of amenable type. Employing the notion of dimension expanders, we give a tangible example of a family of modules for (generalised) Kronecker quivers over arbitrary fields that preclude their amenability. This result is then extended to hereditary wild and strictly wild algebras. Finally, a weak notion of amenable representation type is used to show that no finitely controlled wild algebra over an algebraically closed field is of amenable type.