# Tagged Questions

In mathematics, especially in geometry and group theory, a lattice in $\mathbb{R}^n$ is a subgroup of $\mathbb{R}^n$ which is isomorphic to $\mathbb{Z}^n$, and which spans the real vector space $\mathbb{R}^n$. In other words, for any basis of $\mathbb{R}^n$, the subgroup of all linear combinations ...

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### I am looking for an introduction to hyperbolic surfaces as a quotient of the upper half plane by lattices.

I keep coming across results of the form: If we take the quotient of the upper half plane by a Fuchsian group with this property, we get a surface with that property (cusps, funnels, in/finite area, .....
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### What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?

I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices. Is there an intuitive way to think about it, ...
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### Easy examples of non-arithmetic lattices

I'm starting to look a bit more at discrete subgroups of Lie groups, particularly lattices. A lot is written about arithmetic lattices of Lie groups, and examples abound. It appears that much less is ...
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### What is the Haar measures on $SL(2, R$ And $SL(2,R) / SL(2, Z)$?

How does one parametrize those spaces in order to do integration over them? What's a good reference for doing integral a with Haar measures over matrix groups?
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### Questions about definition of edges in affine building.

I have a question about about edges in a building in the book of expander graphs by Alexander Lubotzky, page 69. We know that if $L_1' \subseteq L_2'$ and $[L_2' : L_1'] = p$, then there is an edge ...
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### Classification of 6-D Nilmanifolds

I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...