1
vote
1answer
98 views
General and translational Birkhoff lattices. Equational classes.
By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to b …
1
vote
0answers
72 views
A question on the theorem of Minkowski-Hlawka
The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. …
8
votes
4answers
466 views
Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of typ …
7
votes
6answers
891 views
Giving $Top(X,Y)$ an appropriate topology
I am not sure if its OK to ask this question here.
Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$.
Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a fu …
7
votes
1answer
136 views
Is the group of integer points of ${\rm SO}(n,1)$ maximal?
That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite inde …
5
votes
2answers
374 views
Area of a lattice polygon in terms of its width
Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M) …
1
vote
1answer
79 views
Lattice basis with Gram-Schmidt vectors of increasing length
Let $\Lambda$ be an $n$-dimensional lattice in $\mathbb R^n$ and let $\cal B$ be the set of all bases that generate $\Lambda$. For a basis
$\mathbf{B}=[\mathbf{b}_1, ... ,\mathbf{ …
4
votes
1answer
219 views
A question of compactness in the geometry of numbers
Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants …
0
votes
0answers
74 views
Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.
I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric …
5
votes
4answers
402 views
Applications of n-dimensional crystallographic groups
I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides …
0
votes
0answers
87 views
Minkowski’s successive minima: A quantity not much larger than det(L)^(1/n) and not much smaller than λ_n(L)?
Let $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ be $n$ linearly independent vectors in an $n$-dimensional lattice $\Lambda$ in $\mathbb{R}^n$ and let $\mathbf{v}^*_1 ,\mathbf{v …
10
votes
2answers
434 views
discrete subgroups of Lie groups and actions on homogeneous spaces
Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ …
4
votes
2answers
411 views
Property of lattices in Lie groups
Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each …
4
votes
1answer
111 views
Lattice in motion group
Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group
$$
G={\mathbb R}^d\rtimes O(d).
$$
Let $\phi:G\to O(d)$ the projection homomorphism.
Is it true that $\p …
0
votes
1answer
76 views
orthogonality in a lattice
Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subs …

