Questions tagged [arithmetic-groups]

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1answer
109 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 ...
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0answers
68 views

Small pants in arithmetic hyperbolic surfaces of high degree

Does the following statement hold: Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface  $\Gamma$ ...
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0answers
43 views

Commensurability of arithmetic, irreducible, nonuniform lattices

Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
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0answers
89 views

Definition of reducible lattice

I am reading Raghunathan's book on discrete subgroups of Lie groups. In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
5
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0answers
106 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
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0answers
44 views

finite subgroups of discrete arithmetic groups

Let K be a totally real multi-quadratic fields and let $\mathcal{O}$ be its ring of integers. I would like to compute the orders of the finite subgroups of the discrete group $\mathrm{SL}_{2}(\mathcal{...
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133 views

Matrix factorizations over $GL_2$ of a real quadratic ring of integers

tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition, the KAN decomposition, the Schur triangular form, etc). Those ...
4
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1answer
183 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 ...
4
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1answer
163 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 $\...
4
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1answer
134 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 ...
4
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1answer
186 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^...
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68 views

A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$. From "A note on generators for arithmetic subgroups of algebraic groups" by ...
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68 views

Congruence subgroup property for unipotent groups over $\mathbb Z[\alpha]$

Let $\alpha$ be an algebraic element over $\mathbb Z$ and mark $\overline {\mathbb Z}=\mathbb Z[\alpha]$. If $Λ$ is a subgroup of $SL_n(\overline {\mathbb Z})$ and $U$ an unipotent subgroup of $SL_n(\...
1
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1answer
132 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 ...
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115 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 $\...
5
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0answers
103 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 ...
7
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1answer
286 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 ...
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0answers
515 views

Pointwise stabilizer of an apartment of the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$

Denote by $X$ the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$. Let $\Sigma$ be the fundamental apartment of $X$. Let $\Gamma=\mathrm{SL}_n(\mathbb{Z}[\frac{1}{p}])$. We can prove that the ...
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0answers
84 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 ...
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0answers
202 views

Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$

I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...
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0answers
60 views

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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0answers
56 views

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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0answers
110 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{...
2
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1answer
147 views

When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?

In the paper p-adic L-functions and p-adic periods of modular forms, Greenberg/Stevens assert that if $\sigma_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double ...
4
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1answer
278 views

fundamental domains in H^2 containing large balls

I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
4
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1answer
114 views

On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$. As usual, $O(V)$ denotes the orthogonal ...
5
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2answers
191 views

Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty: Specifically, ...
9
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1answer
303 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 ...
10
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1answer
342 views

What does the $p$-adic closure of an arithmetic lattice look like?

Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{...
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0answers
193 views

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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0answers
129 views

$\text{PGL}_n(\mathbf{Q}_p)$ 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 $\...
9
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1answer
166 views

Lower bounds for the top rational cohomology of arithmetic groups

I would like to know what estimates exist for the dimension of $H^d({\rm GL}_2(\mathcal{O}_{K,S}),\mathbb{Q})$ where $\mathcal{O}_{K,S}$ is a ring of $S$-integers in a number field $K$ and $d$ is the ...
1
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2answers
658 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 ...
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0answers
128 views

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(\...
5
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2answers
208 views

integer matrices with non-real spectra

I am interested in infinite order elements $A\in SL(3, {\mathbb Z})$ whose spectra are not contained in ${\mathbb R}$ (i.e. such $A$ has two distinct complex-conjugate eigenvalues which are not roots ...
4
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1answer
209 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}/...
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1answer
324 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 $\Lambda$ ...
8
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0answers
168 views

Non-vanishing of the Borel classes in the cohomology of $SL_n(\mathbb Z)$

The stable real cohomology of $SL_n(\mathbb Z)$ was computed by Borel: it is given by $\mathbb R[z_i\mid i=5,9,13,\cdots]$ with $z_i$ in degree $i$. One may wonder whether the pull back of the stable ...
2
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1answer
316 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?
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2answers
2k views

nonabelian reciprocity law

I heard the following relation in a talk by Peter Scholze. Could someone explain "in a simple way" what is the precise relation between the polynomial $x^4-7x^2-3x+1 $ and the integral homology of the ...
4
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0answers
232 views

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 ...
13
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1answer
333 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. ...
1
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1answer
178 views

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(\...
10
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1answer
356 views

Is there a notion of hyperbolicity for number rings?

For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points,...
4
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1answer
242 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 ...
9
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1answer
290 views

Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?

Let $k$ be a global field and let $S$ be a non-empty set of places containing all archimedean ones. Suppose $f:G\to H$ is a surjective $k$-morphism of $k$-groups and let $\Gamma\leq G(k)$ be an $S$-...
5
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4answers
470 views

Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not co-compact

Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that (1) $\Gamma$ has finite covolume (2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper ...
19
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1answer
275 views

How many ways can I factor a matrix (over $\mathbb{Z}$)?

Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$. Question 1 How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$? The answer to this question is pretty clearly ...
4
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1answer
147 views

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 ...
5
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1answer
397 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-...