All Questions
Tagged with arithmetic-groups geometric-group-theory
7 questions
19
votes
2
answers
5k
views
Minimal number of generators for $GL(n,\mathbb{Z})$
$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = \begin{...
- 1,312
13
votes
1
answer
910
views
Holomorphic cusp forms and cohomology of GL(2,Z)
Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
- 40.3k
6
votes
2
answers
325
views
How bad is the modular space?
I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$?
Do we know something about its homology or homotopy groups ?
$\mathbb{H}^{3}$ is the hyperbolic ...
- 443
6
votes
1
answer
163
views
Are double cosets of cyclic subgroups separable in a special linear group?
Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set
$$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$
Is $S$ closed in the profinite topology on
$\mathrm{SL}_3(\mathbb{...
- 11.3k
5
votes
2
answers
409
views
Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$
I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:
Specifically, ...
- 6,025
4
votes
1
answer
390
views
When does a subgroup of $\operatorname{GL}(n, \mathbb Q)$ have a bounded fundamental domain on $\mathbb R^n$?
$\DeclareMathOperator\GL{GL}$Let $G \subset M_{n\times n~}(\mathbb Z)$ be a finitely generated subgroup of $\GL(n,\mathbb Q)$ (i.e. $g\in G$ is an invertible matrix with entries in $\mathbb Z$). Then $...
- 3,472
3
votes
1
answer
236
views
Is there a bound on the rank of finite index subgroup of SL_3(Z)?
Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?
- 11.3k