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If $G$ is a locally compact group and $\Gamma $ is a discrete subgroup such that the quotient $G / \Gamma$ carries a finite left $G$-invariant Haar measure, then we say that $\Gamma$ is a lattice in $G$.

Why are lattices important? Can you give some motivating examples? What are some applications?

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    $\begingroup$ Aren't we meant to define what we are talking about here? There are multiple places where the word "lattice" is used in maths, so I wanted to clarify. $\endgroup$ – Rafael Sep 7 '18 at 11:11
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    $\begingroup$ The question is possibly too broad (and has no definite answer so hopefully will be made cw) but certainly has to do with mathematical research. $\endgroup$ – YCor Sep 7 '18 at 11:58
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    $\begingroup$ @DavidG.Stork I see nothing wrong with questions providing context for the question, provided it is kept to a reasonable length. Furthermore, isn't the definition given in the question the standard meaning for "lattice inside a group"? That is certainly a commonly accepted term well known to people in representation theory, harmonic analysis, ergodic theory, some areas of operator algebras.... $\endgroup$ – Yemon Choi Sep 7 '18 at 15:20
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    $\begingroup$ The first sentence makes this a more pleasant question for me to read, as someone who does not immediately know what a lattice in a group is. $\endgroup$ – mme Sep 7 '18 at 15:26
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    $\begingroup$ @DavidG.Stork: I believe that it is considered good and recommended practice to provide a bit of context to a question. In this case, remember that lattice has at least one other meaning in mathematics, so it is convenient for the reader to know what concept the OP had in mind. $\endgroup$ – Alex M. Sep 7 '18 at 16:27
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I think this is a fine question and I would like to give it a shot.

It is an empirical phenomenon that there is an abundance of lattices in the mathematical nature, and some of these are of fundamental importance (see some examples below). The importance of these groups does not necessarily stem from being lattices, but from their roles in number theory/ geometry/ group theory. However, the fact that they could be represented as lattices gives us ways to approach them and better understand them.

The main philosophy is that, while the study of a discrete group $\Gamma$ might be hard, if $\Gamma$ could be represented as a lattice in an enveloping group $G$, this study could be replaced, to a great extent, by the easier study of the action of $G$ on the space $G/\Gamma$. In the converse direction, a possible way to approach the study of a space $M$ is to present it as $G/\Gamma$ (more generally, $K\backslash G/\Gamma$) where $\Gamma<G$ is a lattice ($K<G$ is compact).

Let me start with the most basic example, the group of integers $\mathbb{Z}$, which is a lattice in $\mathbb{R}$. Since the times of Fourier we all know how powerful is the interplay between the study of $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$. In another direction, you get Dirichlet's approximation theorem as an immediate consequence of the fact that $\mathbb{Z}<\mathbb{R}$ is a lattice.

Generlizing the embedding $\mathbb{Z}<\mathbb{R}$, given a ring of integers $\mathcal{O}$ in a number filed $K$ we can naturally embed $\mathcal{O}$ as a lattice in a certain euclidean space $\mathbb{R}^r\oplus\mathbb{C}^s$, using filed embeddings of $K$. This enables us to use geometric ideas in algebraic number theory proving easily non-trivial results. Further, $K$ itself could be seen as a lattice in $\mathbb{A}$, the adeles, and harmonic analysis on the latter (to be more precise, on the group of ideles) provides valuable information on $K$.

It was proved by Siegle that $\text{SL}_n(\mathbb{Z})$ is a lattice in $\text{SL}_n(\mathbb{R})$. The fact that $\text{SL}_n(\mathbb{Q})$ is a lattice in $\text{SL}_n(\mathbb{A})$ follows easily. Analogues for arbitrary number fields and algebraic groups were proven by Borel and Hare-Chandra, showing that (under some conditions) arithmetic groups are lattices in the aformentioned $\mathbb{R}^r\oplus\mathbb{C}^s$-points of the corresponding algebraic groups and the adelic analogue. Analogues in positive characteristic were provided by Harder.

The presentation of arithmetic groups as lattices enabled the application of a variety of techniques and gave rise to numerous applications. Let me mention here the amazing Margulis' Normal Subgroup Theorem: under some assumptions, every normal subgroup in an arithmetic group is either finite or of finite index. Let me stress that that presenting these groups as lattices is an essential step in the proof of this theorem (though special cases, such as $\text{SL}_n(\mathbb{Z})$ could be treated otherwise). Margulis also use ergodic theoretical techniques to prove the Oppenheim conjecture using such presentations. There are many, many other applications to Diophantine approximation and lattice point counting problems, see for instance this book.

In the realm of computer science, the presentation of arithmetic groups as lattices provide examples and explicit constructions of expander graphs and Ramanujan graphs, as well as higher dimensional analogues. One can also use this to error correction code, see this recent article.

There are also applications of lattices to "abstract" group theory. An excellent example is the construction of Burger and Mozes of a finitely presented simple group (have a look at this text by Caprace). Further simple groups with various properties were constructed by Caprace-Remy, presenting various Kac-Moody groups as lattices.

Let me now turn to applications to geometry. Every finitely generated, torsion free nilpotent group sits naturally as a lattice in its Malcev completion, thus could be seen as a fundamental group of a nilmanifold and we have a complete dictionary between nilmanifolds and nilpotent groups. In fact, it appears that for many natural classes of manifolds the fundamental groups are lattices. The fantastic Thurston Geometrization Conjecture (proven by Perelman) states essentially that any compact 3-manifold could be decomposed into such. Among these pieces, by far the most complicated ones are hyperbolic 3-manifolds. Using in essential way the fact that their fundamental groups are lattices in $\text{SL}_2(\mathbb{C})$ via the work of Kahn and Markovic (see Ian's answer here), Agol and Wise recently gave us a detailed understanding of the geometry of these manifolds. Also in higher dimensions, using the fact that their fundamental groups are lattices, Mostow proved powerful rigidity results regarding compact hyperbolic manifolds, and these were generalized later also to other kinds of symmetric spaces.

There are numerous other applications, but I think I should stop now.

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This question is really too broad, given the ubiquity of lattices in Mathematics. Just to have an idea, you can google "Lattices and $x$", where $x$ is one the following (huge) subjects:

  • Sphere packings
  • Geometry of numbers
  • Complex abelian varieties
  • Locally homogeneous manifolds
  • Criptography and digital communication via algebraic codes (for instance, Golay code).
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    $\begingroup$ Francesco, the body of the question makes it clear that the OP is only referring to lattices in (Lie) groups $\endgroup$ – Yemon Choi Sep 7 '18 at 15:16
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    $\begingroup$ As do the tags chosen by the OP $\endgroup$ – Yemon Choi Sep 7 '18 at 15:22
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    $\begingroup$ @YemonChoi But $\mathbb R^n$ is a locally compact Lie group, so geometry of numbers seems very relevant. For that matter, $(\mathbb Q_p^*)^n$ is a locally compact Lie group (although admittedly not a real Lie group), and its quotients by discrete subgroups are intimately connected to the theory of abelian varieties over $\mathbb Q_p$. $\endgroup$ – Joe Silverman Sep 7 '18 at 15:41
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    $\begingroup$ Yes, the examples in the answer address the OP's question (4 out of 5 being about lattices in abelian groups, especially Euclidean lattices). $\endgroup$ – YCor Sep 8 '18 at 1:10

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