Density ofg-Vector Cones From Triangulated Surfaces

Abstract

We study g-vector cones associated with clusters of cluster algebras defined from a marked surface (S, M ) of rank n. We determine the closure of the union of g-vector cones associated with all clusters. It is equal to R n except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in R n . Our main ingredients are laminations on (S, M ), their shear coordinates and their asymptotic behavior under Dehn twists. As an application, if (S, M ) is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If (S, M ) is a closed surface with exactly one puncture, it has precisely two connected components.

Train track combinatorics and cluster algebras

“…The g-vector fan is complete if and only if the cluster structure (i.e., the mutation class) is of finite. T. Yurikusa studies the class of cluster algebras whose support of the g-vector fan is dense [Yur20,Yur21]. In particular, the cluster algebra of almost every marked surface is a member of this class [Yur20, Theorem 1.2].…”

Complete gentle and special biserial algebras are $g$-tame

“…A main ingredient of our proof of Theorem 1.6 is the asymptotic behavior of g-vectors under Dehn twists. This proof is inspired from the proof of [Yur,Theorem 1.5]. In the forthcoming paper [Aok], this method plays a key role for analyzing the polytope associated with the fan F (A(D)).…”