All Questions
6 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
14
votes
1
answer
502
views
Abelianization of $\mathrm{GL}_n(\mathbb{Z})$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$What is the abelianization of $\GL_n(\mathbb{Z})$? I know the abelianzation of $\GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field and the ...
- 911
4
votes
1
answer
303
views
Automorphisms of products of $GL_n(\mathbb{Z})$ 's
It is a Theorem of Hua y Reiner (1951) that the group or outer automorphisms $Out(GL_n(\mathbb{Z}))$ is either isomorphic to $\mathbb{Z}/2$, if $n$ odd or $n=2$, or to $\mathbb{Z}/2 \times \mathbb{Z}/...
- 419
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
1
vote
1
answer
155
views
Upper bounds for difference of entries between matrices and their inverses in $\mathsf{GL}_k(\mathbb Z)$
Let $a(M)$ be the maximum absolute value of entries of matrix $M\in\mathsf{GL}_k(\mathbb Z)$.
$M^{-1}\in\mathsf{GL}_k(\mathbb Z)$ holds.
What is a good upper bound for $|a(M)-a(M^{-1})|$?
I am ...
- 1,836
1
vote
1
answer
196
views
If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?
If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is ...
- 332