# Lattices inside matrix groups $SL_2(K)$

I am currently a second year undergraduate majoring in math and our university is offering an opportunity for undergraduates to do a project over the summer break. I have spoken to my professor who is offering this particular project, but he says there aren't a lot of easily accessible material.

Here's the project title: Lattices inside matrix groups.

Project description: Let $G$ be a group of matrices such as the special linear group $SL_2(K)$ over a field $K$. An important class of subgroups of $G$ is the class of lattices: these are subgroups of $G$ that are, roughly speaking, "not too big" and "not too small". The aim of the project is to investigate the geometry of the space of all lattices inside $G$. This involves ideas from group theory, graph theory, algebraic geometry and linear algebra. The space of lattices is determined by solutions of certain polynomial equations, so the project will involve a mixture of theoretical ideas and concrete calculations.

So based on what this project is about, would someone be able to recommend to me any books about lattices or resources that would help me to prepare toward such a project?

He also said reading things about topology, differential geometry and lie groups would also be good but it seems way to advanced for me. I haven't even touched analysis yet, though my professor mentions that this abstract algebra paper that I'm doing at the moment is sufficient background to undertake the project.

In addition, as an introduction to lie groups and lie algebra, what resources or books would one recommend in order to prepare oneself for such a project? As far as book goes, I'm thinking either Naive Lie theory or Brian Hall's Lie Groups, Lie algebras and Representations. Any recommendations and experiences with either book would be helpful.

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You should probably read something on Fuchsian groups to get an idea of the extremely rich geometry of subgroups of $SL_2(\mathbb{R})$. For example, look at the beautiful book Indra's pearls by Mumford, Series and Wright. I think Svetlana Katok's book on Fuchsian groups might be at a good level (it requires a bit of complex analysis, and a bit of algebraic number theory in later chapters though). –  t.b. Sep 14 '12 at 11:41
I should have mentioned that this might not immediately help you with your assignment, but it is something you should be aware of in my opinion. Especially chapters 4 and 5 of Katok's book will provide you with numerous explicit examples of lattices. –  t.b. Sep 14 '12 at 18:44
Thanks for letting me know. I'll try to get a hold of these books! –  tcmtan Sep 14 '12 at 22:33
Okay, this hasn't received that much attention, so far. I'll see if I can give you a few more references shortly but I'm not an expert... Good recommendations for books on Lie groups are in this recent thread. I do have one question: your project description gives $SL_2$ as a specific example, and I was addressing only this case. Do you know whether the project will be restricted to $SL_2$ or will lattices e.g. in $SL_n$ for $n \geq 3$ also be considered? The world of lattices is quite different if $n =2$ or $n \geq 3$. –  t.b. Sep 15 '12 at 10:57
Thanks! My professor mentioned $SL_2(\mathbb{C})$ in particular, but I'm not sure whether $n\ge 3$ will be considered. The project is done over a 10 week period, so I guess it depends how much the student can cover during that time? I should note that it's not definite that I'll be undertaking the project as the department is still reviewing applicants and it's quite competitive. Nonetheless, I just want to prepare myself in case I get the chance to do it. But regardless of whether or not I get picked, I'd still like to learn more about this project since it seems very interesting. –  tcmtan Sep 15 '12 at 11:36

This does not address your question directly but you asked about learning of Lie groups from Hall's book. I am using that book right now for a representation theory course and I highly recommend it. Some comments about the book/prerequisites that you may need to know:

Analysis: Being comfortable with topological notions of compactness, connectedness, convergence (this is very heavy in chapter 2) and of course know what a closed set is. In fact, the definition of a matrix Lie group given in Hall is a closed subgroup of $\textrm{GL}_n(\Bbb{C})$. By closed we mean of course with respect to the topology on $\Bbb{C}^{n^2}$ generated by the usual Euclidean metric. Concerning the topology of matrix Lie groups, you will learn that connectedness and path connectedness are equivalent. In fact the proof that a connected matrix Lie group $G$ being always path-connectedness uses the wonderful fact that any $g \in G$ can be written as $\exp{X_1}\ldots\exp{X_n}$ for $X_1,\ldots,X_n \in \mathfrak{g}$. $\mathfrak{g}$ is the Lie algebra of $G$.

Algebraic Topology: This is used in chapter 3, where there is a proof that for a simply connected matrix Lie group $G$, there is a Galois Correspondence thingy between Lie group homomorphisms and Lie algebra homomorphisms. In fact there is a really slick proof of this correspondence relying on a fact on covering spaces that given a simply connected space $X$ (namely $\pi_1(X) = 0$), there are no proper covering spaces of $X$.

Linear Algebra: Basic facts about complexifications of vector spaces (e.g. to complexify a real vector space $V$ you do the usual extension of scalars process of taking $V \otimes_\Bbb{R} \Bbb{C}$); in fact you should be familiar with the following facts:

1. Given a vector space $V$ over a field $F$, an operator $T$ on $V$ is nilpotent iff there is a basis for $V$ such that $T$ in that basis is upper triangular with all zeros on the diagonal.

2. Given an operator $T$ on a complex vector space, there is a basis for $V$ such that $T$ in that basis is upper triangular. Note that to prove fact 1. you cannot use fact 2. because 1. is true over any field (to my knowledge of characteristic 0).

3. Variations of the Complex Spectral Theorem ( by this I mean the one that extends to beyond self-adjoint operators). This fact is very useful in proving things like the exponential map $\exp : \mathfrak{u}(n) \to U(n)$ is surjective.

4. Familiarity with the Gram - Schmidt Process: This is actually very important because you can use it to prove that $\textrm{SL}_n(\Bbb{R})$ is connected. In fact it gives you your Iwasawa decomposition of $\textrm{GL}_n(\Bbb{C})$. This in turn shows that $\textrm{GL}_n(\Bbb{C})$ is homotopy equivalent to $\textrm{U}(n)$ and so $\pi_1(\textrm{U}(n)) \cong \pi_1(\textrm{GL}_n(\Bbb{C}))$.

I would say that I have not studied the material that you are talking of but if you want just a general understanding of matrix Lie groups, I think chapters 1 - 3 of Hall is enough. Chapter 4 goes into representation theory and you may not need it for your project. In summary, I highly recommend Hall as an introductory text for matrix Lie groups because it is not necessary to know say what a manifold or a smooth immersion is in order to enjoy the book.

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Thanks! Excellent post! This will help greatly with respect to prerequisites. –  tcmtan Sep 17 '12 at 10:48