Cluster algebras and triangulated orbifolds

TL;DR

This paper extends the geometric realization of non-exceptional mutation-finite cluster algebras to skew-symmetrizable cases using hyperbolic orbifolds, establishing key properties like positivity and sign-coherence.

Contribution

It generalizes the theory of Fomin and Thurston to skew-symmetrizable cluster algebras and provides new geometric interpretations and properties.

Findings

Cluster variables are renormalized lambda lengths on hyperbolic orbifolds.

Computed growth rates of these cluster algebras.

Proved positivity of Laurent expansions and sign-coherence of c-vectors.

Abstract

We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, provide positivity of Laurent expansions of cluster variables, and prove sign-coherence of c-vectors.