$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. The resulting skew-group algebra $R \ast G$ is $3$-Calabi-Yau, and can sometimes be endowed with the structure of a $3$-preprojective algebra. However, not every such $R \ast G$ admits such a structure. The finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ are classified into types (A) to (L). We consider the groups $G$ of types (C) and (D) and determine for each such group whether the algebra $R \ast G$ admits a $3$-preprojective cut, that is a $3$-preprojective structure arising from a grading of the McKay quiver of $G$. We show that the algebra $R \ast G$ admits a $3$-preprojective cut if and only if $9 \mid |G|$. Our proof is constructive and yields a description of the involved $2$-representation infinite algebras. This is based on the semi-direct decomposition $G \simeq N \rtimes K$ for an abelian group $N$, and we show that the existence of a $3$-preprojective structure on $R \ast G$ is essentially determined by the existence of one on $R \ast N$. This provides new classes of $2$-representation infinite algebras, and we discuss some $2$-Auslander-Platzeck-Reiten tilts. Along the way, we give a detailed description of the involved groups and their McKay quivers by iteratively applying skew-group constructions.