Questions tagged [arithmetic-groups]
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148 questions
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How does $SL_2(\mathbb Z[\sqrt 2])$ embed into $SL(2,\mathbb R)\times SL(2,\mathbb R)$, and what are the centralizers in $Mat(4\times 4,\mathbb Z)$?
I am trying to learn arithmetic groups, from a dynamical point of view. These questions (maybe silly) come to my mind, but I do not know the answer.
How does $SL_2(\mathbb Z[\sqrt 2])$ embed into $SL(...
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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 ...
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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 ...
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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 ...
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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 ...
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Equivalence of Lie theoretic definitions of an arithmetic lattice
In Margulis' book, there are actually two definitions of arithmetic lattices:
If $\mathbf{G}$ is a connected semisimple algebraic $\mathbb{R}$-group, then a lattice $\Gamma \subset \mathbf{G}(\mathbb{...
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On equality of two quotients of a congruence subgroup
Related question: Non-torsion part of the abelianisation of congruence subgroups
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
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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 $...
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$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property
Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...
user72293
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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-...
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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\...
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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 ...
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Subgroup of $SL_2(O)$ with nice fundamental domain in complex upper half-plane
Let $O$ be the ring of $S$-integers in a real quadratic number field. Let $G$ be an $S$-arithmetic subgroup of $SL_2(O)$ whose intersection with $SL_2(\mathbb Z)$ is not of finite index in $SL_2(\...
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The action of an S-arithmetic group on the hyperbolic plane
I have a really quick question. I am interested in $G=SL_2(\mathbb{Z}[1/p_1,...,1/p_n])$, where $p_1$,..., $p_n$ are prime numbers. Since $G$ is a subgroup of $SL_2(\mathbb{R})$, it acts in the ...
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Hecke eigensystem in cohomology vs. compactly supported cohomology
What follows is a question that's probably well-known to experts, but I haven't been able to find a reference.
Let $\mathrm G$ be a connected, semisimple $\mathbb Q$-group. Let $K \subset \mathrm G(\...
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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 ...
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Geodesic symmetry of a locally symmetric space
Let $M = \Gamma \backslash G/K$ be a Riemannian locally symmetric space, where $G$ is a connected semisimple Lie group of rank at least $2$, $K$ its maximal compact subgroup and $\Gamma < G$ an ...
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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
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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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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(\...
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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(\...
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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 ...
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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 $...
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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$ $|$ $x,y \in \mathbb{Z}$}...
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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 ...
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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 ...
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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 ...
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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)...
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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 $g$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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/...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 \...
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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 ...
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