All Questions
7 questions
7
votes
1
answer
467
views
Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
- 332
6
votes
1
answer
214
views
Are the integer matrices in SO(3,2) "boundedly generated"?
Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.
(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
- 11.3k
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 ...
- 2,050
3
votes
2
answers
1k
views
Examples of groups for which Margulis superrigidity theorem applies
I am not an expert at all in the subject of Lie groups, lattices, arithmetic groups and rigidity. But, lately I am interested in Margulis superrigidity theorem, which in most versions can be stated as ...
- 419
1
vote
0
answers
151
views
On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$
$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...
- 10.5k
1
vote
0
answers
154
views
Definition of arithmetic subgroups of Lie groups
In Maclachlan-Reid we can read
Let $G$ be a connected semisimple Lie group with trivial centre and no compact factor. Let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Then $\Gamma$ is ...
- 563
0
votes
0
answers
267
views
Definition of reducible lattice
I am reading Raghunathan's book on discrete subgroups of Lie groups.
In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
- 101