A general group-theoretic lattice is usually defined as something like

A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant finite measure.

The prototypical example of these sets $G = \mathbb{R}^n$, and $\Gamma\cong \mathbb{Z}^n$ a Euclidean lattice. Even in this special case, the study of such lattices can become quite deep, for example

• they are central to the study of optimal (periodic) sphere packings in $\mathbb{R}^n$, and occasionally central to the study of optimal non-periodic sphere packings
• As such, they are central to certain problems in coding theory/wireless communications, and
• they are also central to "lattice-based cryptography", which is a leading candidate for cryptography that is plausibly secure against adversaries with quantum computers ("post-quantum cryptography").

I am mostly interested in the above (namely lattice-based cryptography). I'm always interested to see stronger mathematical tools applied to my (relatively matheamtically mundane) area. Examples include

• The usage of Gaussians on lattices to prove reductions between lattice problems, in particular in establishing a theoretical basis for the hardness of the Learning with Errors problem

• Mixing time bounds for random walks in the Arakelov Class group to give relatively tight worst-case to average-case reductions for lattice problems for lattices in the ring of integers of a number field $\Gamma\subseteq\mathcal{O}_K$

• A more general version of the above, to generalize the above "ideal lattices" to what are known as "module lattices" in the lattice-cryptography literature.

Anyway, when I casually glance through the literature on lattices in arbitrary locally compact groups $G$, I see quite a few results that people seem to think are fundamental, for example

• Mostow's rigidity theorem,
• things like (Super)strong Approximation

I'm sure there are other non-trivial/exciting results. I don't claim to be an expert. My question is the following

Do any of the central results on lattices in arbitrary locally compact groups $G$ have non-trivial consequences for the study of lattices in $G = \mathbb{R}^n$?

Here, "non-trivial" can be interpreted quite generally, though I would be especially interested if these non-trivial facts did not coincide with "standard" non-trivial facts regarding lattices in $\mathbb{R}^n$, say things like Minkowski's theorems, or things like Siegal/Rodgers' integration formula, etc.