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Spectral gap for SL(2,Z)

Does anybody know any simple proof of Selberg's theorem which states that the first positive eigenvalue $\lambda_1$ of the hyperbolic Laplacian operator on $L^2(SL(2,\mathbb{R})/ SL(2,\mathbb{Z}))$ is ...
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1answer
27 views

Scalar multiple of one lattice contained in another

I believe my question boils down to the following question: Given lattices $L$ and $L'$ in $k^{n}$, does there exist $\lambda \in k^{\times}$ so that $\lambda L' \subseteq L$ and $\lambda L' ...
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32 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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1answer
77 views

What is the dual lattice of Kagome lattice?

We know that the dual lattice of a triangular lattice is the honeycomb lattice. What is the dual lattice of Kagome lattice?
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1answer
41 views

Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
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1answer
57 views

Why has $\mathrm{GL}(n, \mathbb{R})/\mathrm{GL}(n, \mathbb{Z})$ infinite co-volume?

The space $X=\mathrm{SL}(n, \mathbb{R})/\mathrm{SL}(n, \mathbb{Z})$ can be identified as the space of unimodular lattices in $\mathbb{R}^n$ and it is well-known that if we take the Haar measure on ...
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16 views

Closed $U$-orbits in the space of plane lattices

Let $G=\mathbf{SL}_n(\mathbf{R})$ and $\Gamma=\mathbf{SL}_n(\mathbf Z)$, so $G/\Gamma$ is the space of unimodular plane lattices. Let $U$ be the upper unipotent group, that is the set of ...
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1answer
29 views

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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31 views

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 ...
2
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1answer
134 views

Example of a discrete uniform subgroup of a Lie group which is not virtually torsion-free

I'm looking for an example, or source of examples, of a discrete uniform subgroup $G$ of a Lie group $\Gamma$, with $G$ not virtually torsion-free. By uniform subgroup I mean that the quotient ...
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30 views

Enlightening explanation of a theorem of Zimmert's

I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper: Zimmert, R. Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
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1answer
107 views

Infinite index subgroups of $SL(3,\mathbf{Z})$

Finite index subgroups of $SL(3,\mathbf{Z})$ are well-know (at least we know that they are congruence subgroups). But I wasn't able to find reference on infinite index subgroups. Does someone knows ...
3
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1answer
154 views

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 ...
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182 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
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97 views

lattices in semisimple Lie groups

I would like to learn more on lattices in semisimple Lie groups, especially their relations with Coxeter groups. Does anyone have suggestions of books that could be useful? Thanks!