is the case for general $(\operatorname{FI}_V \times \operatorname{FI}^{\operatorname{op}})$-modules is fairly technical, and requires results from [@SS2]. As we do not prove this strengthening of Theorem \[sinkexpstab\] in this work, the statement of that theorem will remain unchanged.\

Discrete Morse theory {#dmt}
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The first step in proving that the $\operatorname{FI}_V$-module $H_q(\operatorname{Conf}^{\operatorname{sink}}_{\bullet}(G,V))$ is finitely generated, as well as the first step in proving many topological results about the spaces $\operatorname{Conf}^{\operatorname{sink}}_n(G,V)$, involves placing a cellular model on $\operatorname{Conf}^{\operatorname{sink}}_n(G,V)$. After this is accomplished, we will refine the model using discrete Morse theory. Discrete Morse theory was introduced by Forman as a means of applying classical Morse theory techniques to the study of CW-complexes [@Fo1][@Fo2]. The exposition that follows is based on the exposition in [@R1], which was inspired by the exposition in the works [@FS][@Fo2][@KP].\

\[dmtmaindefs\] Let $Y$ be a CW complex. A **cell** of $Y$ will always refer to an open cell in $Y$. Given a cell $\sigma$ of dimension $i$, we will often write $\sigma^{(i)}$ to indicate that $\sigma$ has dimension $i$. We will write ${\mathcal{K}}$ to denote the set of cells of $Y$ and ${\mathcal{K}}_i$ to denote the set of $i$-cells of $Y$.\
A cell $\tau^{(i)} \subseteq \overline{\sigma^{i+1}}$ is said 