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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](http://en.wikipedia.org/wiki/Fuchsian_group) 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](http://books.google.com/books?id=kC5kdUirHHoC)* by Mumford, Series and Wright. I think Svetlana Katok's book on *[Fuchsian groups](http://books.google.com/books?id=R7uMkmXh1AYC)* might be at a good level (it requires a bit of complex analysis, and a bit of algebraic number theory in later chapters though).2012-09-14
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    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.2012-09-14
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    Thanks for letting me know. I'll try to get a hold of these books!2012-09-14
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    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](http://math.stackexchange.com/q/194419/5363). 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 $n \geq 3$.2012-09-15
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    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.2012-09-15
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    @t.b. I've looked at the thread you linked me to and was wondering, which book would you personally recommend as an introduction to lie groups? Naive Lie Theory or Brian Hall's Book Lie Groups, Lie algebras and Representations? Have you had experiences with either of these books?2012-09-17

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