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 ...

learn more… | top users | synonyms

0
votes
0answers
38 views

Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
3
votes
1answer
80 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
0
votes
0answers
18 views

Unable to find referenced theorem

I've am reading the article "Finitely summable Fredholm modules over higher rank groups and lattices" http://arxiv.org/abs/0806.2759 . Theorem 4.3 here refers to the article Property (T) and rigidity ...
0
votes
0answers
13 views

Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group G by the left action of a maximal lattice 􀀀, i.e. a discrete cocompact subgroup. ...
4
votes
1answer
26 views

Examples of Lattices in $\operatorname{Isom}(H^n)$ for all $n \geq 2$?

I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples. ...
5
votes
2answers
70 views

Does a lattice in $SL_n(\mathbb R)$ which is contained in $SL_n(\mathbb Z)$ have finite index in $SL_n(\mathbb Z)$?

A lattice $H$ in a locally compact group $G$ is a discrete subgroup such that the coset space $G/H$ admits a finite $G$-invariant measure. I have read several places that any lattice H in $SL_n(\...
2
votes
1answer
30 views

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, .....
2
votes
1answer
32 views

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, ...
1
vote
2answers
68 views

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 ...
2
votes
1answer
111 views

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?
3
votes
1answer
34 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' \not\...
1
vote
0answers
75 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 ...
1
vote
1answer
171 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?
1
vote
1answer
58 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 $W_1 \times W_2$? ...
1
vote
1answer
70 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 $\...
1
vote
1answer
33 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 ...
2
votes
0answers
40 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
votes
1answer
185 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 $\...
0
votes
0answers
35 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 ...
1
vote
1answer
169 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
votes
1answer
175 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 ...
17
votes
0answers
228 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 ...
3
votes
1answer
123 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!