# Questions tagged [linear-groups]

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### Centralizer of elements in the upper-triangular matrices

Let $p$ be a prime number and $G=\operatorname{GL}_n ( \mathbb{Z} / p \mathbb{Z} )$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. ...
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### 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 ...
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### 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 ...
331 views

### Non-torsion part of the abelianisation of congruence subgroups

I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead. Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...
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### What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
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### Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements. Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ? I am interested in ...
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### 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 $N$?
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### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
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### Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
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### Maximal order of finite subgroups of $GL(n,Z)$
I am interested in the finite subgroups of $GL(n,Z)$ of maximal order. Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...