# Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

## TL;DR

This paper establishes the decidability of key problems such as flow equivalence and isomorphism in graph C*-algebras and quiver representations, linking deep results in arithmetic groups to operator algebra classification.

## Contribution

It connects results from arithmetic group theory to the decidability of isomorphism and flow equivalence problems in graph C*-algebras and quiver representations, providing new algorithmic insights.

## Key findings

- Decidability of isomorphism and stable isomorphism of unital graph C*-algebras.
- Decidability of flow equivalence for shifts of finite type.
- Decidability of isomorphism of Z-quiver representations.

## Abstract

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and S{\o}rensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).