Mutation-finite quivers with real weights

We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by reflections. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.Comment: 28 pages, many figures; v2: minor correction