All Questions
Tagged with arithmetic-groups gr.group-theory
62 questions
28
votes
1
answer
2k
views
Integer matrices which are not a power
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
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
17
votes
3
answers
4k
views
Generating the symplectic group
The too naive and vague version of my question is the following: given a collection of integer symplectic matrices all of the same size (say 2n by 2n), how can I tell if they generate the full ...
user5117
15
votes
1
answer
462
views
H_3 of SL(n,Z) and SL(n,F_p)
Can anyone tell me what $H_3(SL_n(\mathbb{Z});\mathbb{Z})$ and $H_3(SL_n(\mathbb{F}_p);\mathbb{Z})$ are? It is easy to find references for $H_1$ and $H_2$, but it turns out that I need $H_3$ as well. ...
- 151
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
14
votes
2
answers
2k
views
Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free
My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer:
Open question: Does there exist a finitely generated Zariski-dense torsion-...
- 2,374
12
votes
3
answers
2k
views
Generators for SL_2(R) for rings of integers R
Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?
...
- 270
12
votes
1
answer
404
views
Are finite presentations of arithmetic groups computable?
In this famous paper by Borel and Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, it is proved that, in characterisitic zero, arithmetic groups are finitely presented. I have an extremely ...
- 1,033
10
votes
1
answer
886
views
Definition of an arithmetic subgroup of an algebraic group
I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$.
In Wikipedia you can read:
If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}_n(\...
- 563
10
votes
3
answers
1k
views
Generators for a certain congruence subgroup of SL(n,Z)
I'm looking for a reference (or quick proof) of the following fact. Fix some $n \geq 3$ and some $\ell \geq 2$. Set $\Gamma_n(\ell) = \text{ker}(\text{SL}_n(\mathbb{Z}) \rightarrow \text{SL}_n(\...
- 378
10
votes
0
answers
465
views
A uniform bound for a "true" non-congruence subgroup
Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
- 638
9
votes
1
answer
402
views
Cohomological gap in arithmetic groups
$\DeclareMathOperator\SL{SL}$For the sake of this question, let's say that a group $G$ of finite cohomological dimension $n$ has a cohomological gap if for some $0 < i < n$ there is no subgroup $...
- 423
9
votes
1
answer
384
views
Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I
Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map $\text{SL}_n(\mathcal{O}_K)...
- 91
8
votes
1
answer
339
views
Subgroups of Sp(2g,Z) that map onto all Sp(2g,Z/m)
I stumbled into the following problem. I apologize for being a bit naive.
For $g\geq 3$, consider the group $\mathrm{Sp}(2g,\mathbb{Z})$ of symplectic square matrices of order $2g$ with integral ...
8
votes
2
answers
336
views
What is the most efficient way to factor a matrix into a given set of generators?
I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general ...
- 2,436
8
votes
0
answers
201
views
Monodromy groups that are profinitely dense in Sp(2g,Z)
$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
7
votes
2
answers
327
views
Residual $p$-finiteness of principal congruence subgroups
Let $\Gamma(N)$ be the principal congruence subgroup of level $N$ in $\mathrm{SL}_n(\mathbf{Z})$, where $n\geq 3$. Then $\Gamma(N)$ is residually $p$-finite for all primes $p$ dividing $N$.
Can $\...
- 115
7
votes
2
answers
442
views
Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$
The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for $...
- 10.9k
7
votes
1
answer
465
views
Abelianization of Hilbert modular group
Let $d>0$ be a square free positive integer and let $\mathcal{O}_d$ be the ring of integers in $\mathbb{Q}[\sqrt{d}]$. What is the abelianization of the Hilbert modular group $\text{SL}_2(\mathcal{...
- 71
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
7
votes
1
answer
397
views
Cohomology of certain arithmetic groups
This is a question on literature about cohomology of arithmetic groups.
Let $M$ denote a quaternion algebra over $\mathbb Q$ and assume it is non-split over $\mathbb R$. Fix a maximal order $\Lambda$ ...
user1688
6
votes
1
answer
392
views
Free abelian subgroups of $\mathrm{SL}_n(\mathbb{Z})$
Does anybody know what is the biggest $r$ such that $\mathbb{Z}^r$ is isomorphic to a subgroup of $\mathrm{SL}_n(\mathbb{Z})$?
It cannot be bigger that the virtual cohomological dimension of $\...
- 419
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
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}\}.$$
Is $S$ closed in the profinite topology on
$\mathrm{SL}_3(\mathbb{...
- 11.3k
6
votes
1
answer
252
views
A result of Borel on extensions of arithmetic groups
A famous result of Sullivan (closely related to work of Wilkerson) says that the group of isotopy classes of diffeomorphisms of a simply-connected closed smooth manifold of dimension $\geq 5$ is ...
- 8,167
6
votes
0
answers
202
views
Index of subgroups of $\mathrm{Sp}(4,\mathbb{Z})$ conjugate in $\mathrm{GL}(4,\mathbb{Q})$
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($...
- 141
6
votes
0
answers
365
views
Cohomology of $\operatorname{SL}_n(\mathbb Z)$ with coefficients
Let $K$ be a number field, which can be $\mathbb Q(\zeta)$ with $\zeta$ a root of unity if that helps, and let $\operatorname{SL}_n(\mathbb{Z})$ act on $(K^\times)^n:=K^\times\times\cdots\times K^\...
- 47.3k
6
votes
0
answers
164
views
Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?
The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...
- 8,167
5
votes
1
answer
257
views
Is SL_n of an order in a number ring finite-index in SL_n of the number ring?
Let $\mathcal{O}$ be the ring of integers in an algebraic number field and let $R \subset \mathcal{O}$ be an order. For instance, we might have $\mathcal{O} = \{\text{$x+i y$ $|$ $x,y \in \mathbb{Z}$}...
- 53
5
votes
3
answers
448
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
5
votes
1
answer
133
views
Normal closure of $e_{12}$ in the congruence subgroup $\Gamma_1(p)\subset \mathrm{SL}_2(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let
$$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...
- 155
5
votes
1
answer
429
views
Cohomology of linear algebraic groups
Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...
user39380
5
votes
1
answer
391
views
Normalizers in arithmetic groups
This is a question about the class of arithmetic groups. I am using the definition in Serre's survey: $\Gamma$ is arithmetic if it can be embedded into $G_\mathbb{Q}$ for some algebraic subgroup $G \...
- 8,167
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
5
votes
0
answers
267
views
Generating congruence subgroups of SL_n over totally imaginary number rings
Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...
- 51
4
votes
2
answers
207
views
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
- 43
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
4
votes
1
answer
274
views
Subgroups of $Sp(2n,\mathbb{R})$ between $Sp(2n,\mathbb{Z})$ and some arithmetic group
The fantastic answers to my previous question Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup led me to the following question.
Let $O_K$ be the ring of ...
- 195
4
votes
1
answer
291
views
Unitary representations of lattices
Let $G$ be a simple linear group over a non-archimedean local field $F$.
Assume that the split-rank over $F$ is at least 2.
Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated ...
user130903
4
votes
1
answer
218
views
Commensurator of a subgroup of matrices
Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
- 521
4
votes
0
answers
238
views
What is known about the cohomology of the U-duality group?
$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie
group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
- 6,881
4
votes
0
answers
149
views
Generators of special linear group of a projective module over a number ring
Let $\mathcal{O}$ be the ring of integers in an algebraic number field $K$ and let $M$ be a rank-$n$ projective $\mathcal{O}$-module. By definition, this means that $M \otimes K \cong K^n$, so the ...
- 41
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
3
votes
1
answer
159
views
Cohomology of cocompact lattices in hyperbolic spaces
I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
- 131
3
votes
1
answer
220
views
Abelianizations of arithmetic Fuchsian groups
Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group
\begin{equation*}
\Gamma=\left\{\begin{bmatrix}
k+\sqrt{a}l &...
3
votes
1
answer
402
views
Conjugacy of torsion subgroups in Gl(n, Z) for small n [duplicate]
Have the conjugacy classes of the torsion subgroups of Gl(n, Z) been determined for small n (say, n<=6)? In general, can much be said about the torsion subgroup?
- 31
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
3
votes
0
answers
103
views
Colimits in cohomology of profinite arithmetic groups
Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...
- 133
3
votes
0
answers
110
views
Is this a lattice?
Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...
user130903
2
votes
1
answer
425
views
Is $SL_n(\mathbb{Z}_p)$ virtually torsion free?
If so, is there a way to conclude this from Malcev's theorem?
In general, what is known about virtually torsion freeness of non-finitely generated linear groups?
user100742