Universal geometric cluster algebras from surfaces

TL;DR

This paper studies universal geometric cluster algebras from surfaces, focusing on properties like the Curve Separation and Null Tangle, and constructs related fans and coefficients for various surface types.

Contribution

It broadens the definition of geometric cluster algebras, proves key properties for surfaces, and constructs universal coefficients and fans for most marked surfaces.

Findings

Proved the Curve Separation Property for all but once-punctured surfaces without boundary.

Established the Null Tangle Property for a smaller class of surfaces.

Constructed the rational part of the mutation fan for these surfaces.

Abstract

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan F_B called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of F_B for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces.