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.