Cluster algebras and triangulated surfaces. Part I: Cluster complexes

Abstract

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the surface, and determine its homotopy type and its growth rate.

Figures (32)

• Figure 1: Self-folded ideal triangle
• Figure 2: Ideal triangulations of a once-punctured triangle
• Figure 3: The arc complex of a hexagon (solid lines)
• Figure 4: The arc complex of a once-punctured triangle
• Figure 5: Arc complex $\Delta^{\space\circ}(\mathbf{S},\mathbf{M})$ for an annulus of type $\widetilde{A}(2,1)$

Theorems & Definitions (144)

• Definition 2.1: Bordered surfaces with marked points
• Definition 2.2: Arcs
• Proposition 2.3: e.g., fg-dual-teich
• Definition 2.4: Compatibility of arcs
• Proposition 2.5
• Definition 2.6: Ideal triangulations; see, e.g., ivanov
• Example 2.7
• Remark 2.8
• Remark 2.9
• Proposition 2.10