Bounding the covolume of lattices in products

We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is non-compact and every closed normal subgroup of $G_i$ is discrete or cocompact (e.g. $G_i$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\Gamma$ with dense projections is finite. The same result holds if $\Gamma$ is non-uniform, provided $G$ has Kazhdan's property (T). We show that for any compact subset $K \subset G$, the collection of discrete subgroups $\Gamma \leq G$ with $G = \Gamma K$ and dense projections is uniformly discrete, hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $Aut(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.