Questions tagged [arithmetic-groups]
The arithmetic-groups tag has no usage guidance.
148 questions
5
votes
1
answer
599
views
If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?
Let $G$ be an absolutely simple simply connected and connected algebraic group defined over a global field $k$ with ring of integers $\mathcal{O}$. Fix an embedding of $G$ into $GL_n$. Given $v$ a non-...
- 141
5
votes
1
answer
612
views
Is the automorphism group of a Calabi-Yau variety an arithmetic group
Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a ...
- 193
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
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
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
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
4
votes
1
answer
354
views
Volume of arithmetic quotients of symmetric spaces
Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...
- 685
10
votes
3
answers
581
views
Is there a generalization of the "characteristic polynomial" to other split/quasi-split algebraic groups?
Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...
- 1,453
1
vote
0
answers
162
views
Cohomology of discrete group with compact support
This is closely related to a previous question on the topic, but hopefully adds some motivation.
Let $G_{/\mathbf Q}$ be a semisimple group, $K\subset G(\mathbf R)$ a maximal compact subgroup, and $...
- 5,759
3
votes
2
answers
752
views
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Let $G\subseteq GL(n)$ be a linear algebraic group, and let $G({\Bbb Q}_p)\subseteq GL(V)$ act on a ${\Bbb Q}_p$-vector space V of finite dimension.
Consider the action of $G$ on abelian subgroups $L\...
- 1,299
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
3
votes
1
answer
294
views
Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$
Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...
- 31
12
votes
0
answers
287
views
Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?
Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...
- 181
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
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
1
vote
0
answers
140
views
symmetric theta structures and arithmetic subgroups
A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface.
Now,...
- 3,779
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
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
3
votes
3
answers
544
views
Non existence of cyclic infinite linear algebraic groups
Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of $...
- 7,722
11
votes
1
answer
565
views
What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
- 3,186
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
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
3
votes
0
answers
342
views
Discussion of specific arithmetic triangle groups?
Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
- 16.6k
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
7
votes
1
answer
640
views
Characterisation of Q-rank 1
I'm looking for a reference and/or the original source for the following fact:
An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
- 10.4k
2
votes
1
answer
97
views
Action of GL(2,O_k) on 1d subspaces of (O_k)^2
Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k \...
- 23
1
vote
0
answers
190
views
Compactifications of group schemes
Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
- 41
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
8
votes
1
answer
280
views
Is the group of integer points of ${\rm SO}(n,1)$ maximal?
That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite index)? if such a ...
- 353
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 $\Gamma_n(\ell) = \text{ker}(\text{SL}_n(\mathbb{Z}) \rightarrow \text{SL}_n(\...
- 378
1
vote
1
answer
504
views
Siegel set in SO(n,1) modulo integer points?
I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of
$K A_\tau N_c$, where $N_c$ is compact ...
- 353
2
votes
1
answer
533
views
S-arithmetic subgroup question
I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:
Let $G$ be a connected ...
- 47
7
votes
0
answers
472
views
subgroups of higher rank lattices
This is related to the question $G=\langle a\rangle H$ for subgroup $H$ raised a few days ago. Suppose $\Gamma $ is a higher rank lattice (for example, $SL_3({\mathbb Z})$). As Misha says in his ...
- 11.2k
6
votes
1
answer
364
views
2 generated arithmetic groups
Suppose $G({\mathbb Z})$ is a higher rank non-cocompact arithmetic group (e.g. $SL_n({\mathbb Z})$ with $n\geq 3$, or $Sp_{2g}({\mathbb Z})$ with $g\geq 2$). I have seen a result (http://arxiv.org/abs/...
- 11.2k
11
votes
2
answers
3k
views
Why are $S$-arithmetic groups interesting?
Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.
Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
- 349
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
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
4
votes
1
answer
408
views
Genus of arithmetic surface groups
It is known that for each genus, only finitely many points in the moduli space of hyperbolic genus g surfaces are arithmetic. I'm wondering if an existence result is known: for which g do we have ...
- 41
1
vote
1
answer
476
views
Fixed points of the Borel-Serre compactification
Let $\Gamma$ be an arithmetic group and $X$ its symmetric space. Borel-Serre constructed a space $\bar{X} \supset X$ such that $\bar{X}/\Gamma$ is a compactification of $X/\Gamma$ [Corners and ...
- 16.2k
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
1
vote
2
answers
627
views
Zariski density of conjugates of subgroups by arithmetic subgroups?
Let $G$ be a linear algebraic $\mathbb{Q}$-group, which is assumed to be connected, $\mathbb{Q}$-simple, and of adjoint type, such that the Lie group $G(\mathbb{R})$ has no compact factor defined over ...
- 313
2
votes
2
answers
2k
views
arithmetic groups VS. Zariski dense discrete subgroups?
Assume that $G$ is a semi-simple linear algebraic group defined over $\mathbb{Q}$, which is $\mathbb{Q}$-simple, and that $G(\mathbb(R)$ is non-compact, without $\mathbb{R}$-factors of rank 1. Then by ...
- 1,305
4
votes
0
answers
249
views
links and interactions between different approaches to (super-)rigidity
By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...
- 1,305
7
votes
1
answer
2k
views
discrete subgroups in p-adic Lie groups?
It is known, from the works of G.Margulis, etc. that lattices in semi-simple real (algebraic) groups are "often" arithmetic subgroups, as long as the split rank is high enough. Here by a lattice in a ...
- 1,305
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
3
votes
1
answer
1k
views
Quotients of unipotent groups
Let $U (\mathbf{R})$ be the standard unipotent subgroup of $SL(3, \mathbf{R})$. So $U(\mathbf{R})$ is the group of 3 by 3 upper triangular matrices with 1s on the diagonal. I am interested in the ...
- 741
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
4
answers
1k
views
cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
- 4,265