Wild Kronecker quivers and amenability

Abstract

We apply the notion of hyperfinite families of modules to the wild path algebras of generalised Kronecker quivers $k\Theta (d)$. While the preprojective and postinjective component are hyperfinite, we show the existence of a family of non-hyperfinite modules in the regular component for some $d$. Making use of dimension expanders to achieve this, our construction is more explicit than previous results. From this it follows that no finitely controlled wild algebra is of amenable representation type.

arXiv comment18 pages, 2 figures. Updated version fixes some typos, improves some proofs and adds a new section dealing with the non-amenability of controlled wild algebras