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Abstract In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $$B$$, we construct a polygon $$\mathcal {S}$$ with a checkerboard pattern in its interior, which defines a category $$\text {Diag}(\mathcal {S})$$. The indecomposable objects of $$\text {Diag}(\mathcal {S})$$ are the 2-diagonals in $$\mathcal {S}$$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $$\text {Diag}(\mathcal {S})$$ is equivalent to the stable syzygy category of the algebra $$B$$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $$\mathbb {A}$$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $$\mathcal {S}$$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver. 

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichmüller theory to define a super version of the cluster algebra of type A and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type A. 

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner’s lambda lengths and SL2-tilings. In particular, we prove a three-dimensional version of the Ptolemy relation, and generalise results of Short to classify tame SL2-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph. 

A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning that every arrow is contained in a perfect matching; this is not true for general surfaces. Strong consistency is defined to require weak consistency as well as nondegeneracy. We prove that the completed as well as the noncompleted dimer algebra of a strongly consistent dimer model are bimodule internally 3-Calabi-Yau with respect to their boundary idempotents. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of suitable dimer models categorifies the cluster algebra given by their underlying quiver. We provide additional consequences of weak and strong consistency, including that one may reduce a strongly consistent dimer model by removing digons and that consistency behaves well under taking dimer submodels.