Wild Kronecker quivers and amenability

TL;DR

This paper investigates the structure of modules over wild Kronecker quivers, showing that certain components are hyperfinite while others are not, leading to conclusions about the non-amenability of some wild algebras.

Contribution

It introduces explicit constructions of non-hyperfinite modules in regular components of wild Kronecker quivers using dimension expanders, advancing understanding of amenability in representation theory.

Findings

Preprojective and postinjective components are hyperfinite.

Existence of non-hyperfinite modules in regular components for some d.

No finitely controlled wild algebra is of amenable representation type.

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.