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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.

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  • $\begingroup$ I remember that the Benoist-Quint results on random walks in semisimple Lie groups modulo lattices has consequences on random walks on the torus. But I have nothing more precise in mind. $\endgroup$
    – YCor
    Commented Jul 18, 2023 at 21:02
  • $\begingroup$ Isn't studying lattices in $\mathbb{R}^{n}$ basically the same as studying the space $PSL_{n}(\mathbb{R})/PSL_{n}(\mathbb{Z})$? $\endgroup$
    – Asaf
    Commented Jul 19, 2023 at 1:05
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    $\begingroup$ @asaf generally people instead describe it as the double coset space $O(n)\backslash \mathsf{GL}_n(\mathbb{R})/\mathsf{SL}_n(\mathbb{Z})$. This viewpoint is key to some of the non-trivial lattice facts I mentioned, namely the definition of Siegel measure and the Siegel integration formula. If you know of other consequences of viewing the space of all lattices of this form (or of your form) I would be interested in hearing them. $\endgroup$ Commented Jul 19, 2023 at 4:54
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    $\begingroup$ @Mark, my form assumes unimodular lattices. The $O(n)$ part of yours assumes rotation invariance, which is generally not needed. Anyhow, first and foremost, this presentation allows one to discuss a "generic lattice" as one have a finite measure. Plainly speaking, "all" lattices in $PSL_{n}(\mathbb{R})$, for $n\geq 3$, would be "essentially" $PSL_{n}(\mathbb{Z})$, so I don't understand which kinds of result you are after. Certainly any equidistribution theorem would tell you that if you start with a given lattice and modify it in a prescribed manner, you get a random lattice. $\endgroup$
    – Asaf
    Commented Jul 19, 2023 at 5:15
  • $\begingroup$ Check en.wikipedia.org/wiki/Oppenheim_conjecture to see if it is an example of what you are looking for. $\endgroup$ Commented Jul 19, 2023 at 13:37

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