All Questions
Tagged with lattices gt.geometric-topology
17 questions
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 12.9k
5
votes
1
answer
356
views
Lattice generated by parabolics
Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
- 28.2k
3
votes
1
answer
158
views
Cohomology of cocompact lattices in hyperbolic spaces
I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
- 63.9k
14
votes
1
answer
638
views
How do we know there are no more Deligne–Mostow/Thurston lattices?
In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used ...
- 7,226
4
votes
1
answer
131
views
Inheritance of arithmeticity properties in orbifold strata
Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
- 12.3k
11
votes
2
answers
1k
views
Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
- 28.2k
8
votes
1
answer
320
views
Are the determinants of a lattice discrete?
Let $\Lambda\subset \mathbb{R}^4$ be a lattice. We identify $\mathbb{R}^4$ with the space $M_2(\mathbb{R})$ of $2\times 2$ matrices over $\mathbb R$. It then is is clear that the set
$$
\det(\Lambda)=\...
- 28.2k
9
votes
1
answer
444
views
Compact flat orientable 3 manifolds and mapping tori
There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are ...
- 4,081
3
votes
1
answer
133
views
Lattice-like structure with maximum spacing between vertices
I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I ...
- 5,429
2
votes
1
answer
133
views
Do discrete embeddings of surface groups not necessarily carry an embedding of SL_2?
We can get arithmetic lattices isomorphic to free groups in $\mathrm{SL}_2\mathbb{R}$, so in general we can’t expect homomorphisms of lattices into semisimple Lie groups to say much about $\mathrm{SL}...
- 63.9k
9
votes
1
answer
293
views
Euler characteristic with compact support of spaces of Euclidean lattices
Has the Euler characteristic with compact support of $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ been computed ? References? Thanks.
- 68.9k
5
votes
0
answers
150
views
Lattices of minimal covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$
What are the (uniform/non-uniform) irreducible lattices of minimal (or even small) covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$?
Context: Such a lattice will ...
- 63.9k
11
votes
0
answers
239
views
Representation of the space of lattices in $\Bbb R^n$
The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
- 63.9k
1
vote
0
answers
254
views
Defining a notion of “volume of its lattice” for non-rational subspaces
Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...
- 63.9k
7
votes
0
answers
172
views
Subgroups of $\mathbb{Z}^{n}$ with rotational symmetries
Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$
is
$$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\...
- 63.9k
6
votes
2
answers
805
views
Intuition behind the definition of the Siegel-Eichler transformation
Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.
Let $X$ be an ...
5
votes
3
answers
1k
views
Matrices generating non-discrete subgroups of SL(2,R)
Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
- 31.2k