30
votes
Nonabelian reciprocity law
Will Sawin's answer is perfectly correct, but I wanted to add some further perspective.
What Peter Scholze has proven is a profound generalization of (one half) of Class Field Theory. The particular ...
26
votes
Integer matrices which are not a power
I actually even struggle to find examples of primitives matrices in these groups.
Here is a relatively easy sufficient condition. If $M \in SL_n(\mathbb{Z})$ is the $k^{th}$ power of some other ...
- 118k
19
votes
Accepted
Abelianization of $\mathrm{GL}_n(\mathbb{Z})$
For $n=1$ and $n\ge 3$ $\mathrm{SL}_n(\mathbf{Z})$ is perfect and hence the abelianization has order 2, given by the $\pm 1$ determinant.
For $n=2$, $-I_2$ is a commutator so this is the same as the ...
- 63.9k
18
votes
Accepted
Volumes of $\mathrm{SL}_n(K_\mathbb{R})/\mathrm{SL}_n(\mathcal{O}_K)$
$\DeclareMathOperator\SL{SL}$The same argument (due to Siegel, in a classical form, of course), adelized, gives the analogous computation for any number field, and, yes, the corresponding Dedekind ...
- 23k
17
votes
Accepted
H_3 of SL(n,Z) and SL(n,F_p)
Summarizing the comments, the stable ($n\geq 3$) values of $H_3(SL_n;\mathbb Z)$ are
$H_3(SL_\infty(\mathbb Z);\mathbb Z) = \mathbb Z/24$
$H_3(SL_\infty(\mathbb F_q);\mathbb Z) = \mathbb Z/(q^2-1)$
...
15
votes
Accepted
Definition of an arithmetic subgroup of an algebraic group
Write the coordinates $a_{ij}$ of one embedding into $GL_n$ as polynomial functions, defined over $\mathbb Q$, in the coordinates $b_{ij}$ of a different embedding into $GL_n$. We can do this because, ...
- 148k
14
votes
Nonabelian reciprocity law
Peter Scholze explains the same example in more detail in this talk https://youtu.be/cFdm0B9KLcQ?t=48m and gives even more detail here https://youtu.be/gI3Ta73yVuo?t=23m55s.
In brief, the ...
- 148k
13
votes
Is $SL_n(\mathbb{Z}_p)$ virtually torsion free?
$SL_n(\mathbb{Z}_p)$ is virtually torsion free as it is $p$-adic analytic and therefore contains a uniformly powerful open subgroup.
- 5,450
12
votes
Subgroups of Sp(2g,Z) that map onto all Sp(2g,Z/m)
The answer to your question as to whether $G=Sp_{2g}({\mathbb Z})$ is NO. $Sp_{2g}({\mathbb Z})$ contains a pro-finitely dense FREE subgroup (hence has infinite index) which is also finitely ...
- 11.2k
12
votes
Accepted
Arithmetic groups and integral points of integral structures
First question (do non-strictly-arithmetic subgroups exist?):
Any "strictly arithmetic" subgroup in your sense will, in particular, be a congruence subgroup, i.e. the intersection of $G(\...
- 37k
11
votes
Accepted
Covolumes of unit groups of division algebras
First, you need to avoid definite quaternion algebras over $\mathbb{Q}$: in this case, the unit groups are finite, so the index cannot grow with $N$.
With that out of the way, your algebra $D$ ...
- 3,009
9
votes
Accepted
Free abelian subgroups of $\mathrm{SL}_n(\mathbb{Z})$
The answer is $\lfloor n^2/4\rfloor$, namely $m^2=n^2/4$ for even $n=2m$ and $m(m+1)=(n^2-1)/4$ for odd $n=2m+1$.
Indeed, one has a free abelian subgroup of rank $\lfloor n^2/4\rfloor$, consisting of ...
- 63.9k
9
votes
Accepted
Normalizers of the principal congruence subgroups in $\mathrm{GL}(n,\mathbf Q)$
Proposition. For every $n\ge 0$ and $m\ge 1$ the normalizer of $\Gamma_n(m)$ in $\mathrm{GL}_n(\mathbf{Q})$ is $\mathbf{Q}^*\cdot\mathrm{GL}_n(\mathbf{Z})$.
Proof. One inclusion is clear so it is ...
- 63.9k
8
votes
Accepted
What does the $p$-adic closure of an arithmetic lattice look like?
Suppose $G$ is $\mathbb Q$ simple (i.e. has no connected normal algebraic subgroups which are defined over $\mathbb Q$) and is simply connected (i.e. $G(\mathbb C)$ is simply connected). Assume also ...
- 11.2k
8
votes
Accepted
integer matrices with non-real spectra
There are no such pairs.
Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be the eigenvalues of $A$, and $\mu_1,\mu_2,\mu_3$ be those of $B$ (with $\lambda_1,\mu_1\in\mathbb R$). Since the eigenvalues are ...
- 23.7k
8
votes
Accepted
Computing a Commutator Subgroup
With the latest update, indicating that the quadratic form in question is, up to reordering, $\sum x_i x_{-i}$ (here we number the basis elements as $e_1,\ldots,e_5,e_{-5},\ldots,e_{-1}$), the answer ...
- 3,186
8
votes
Accepted
Equidistribution on $\mathrm{SU}_2$
In the article "On the spectral gap for finitely-generated subgroups of SU(2)" by Jean Bourgain and Alex Gamburd (Invent. Math. 171, No. 1, 83-121 (2008)), they show that free subgroups of $...
- 1,058
7
votes
Automorphisms of products of $GL_n(\mathbb{Z})$ 's
Here's a proof of finiteness.
First, it's a general fact that for any centerless directly indecomposable groups $A_1,\dots,A_k$, any automorphism permutes the $A_i$. References are welcome; one (in ...
- 63.9k
7
votes
Accepted
Schur multiplier of a Chevalley group of type $D_5$
According to Theorem 5.3 of the first paper of Mike Stein that you mention, the universal central extension of the Chevalley group of type $D_5$ over the integers is given by the Steinberg generators ...
- 16.2k
6
votes
Accepted
Cohomology of certain arithmetic groups
We can write $G(\mathbb Q_p)/ G(\mathbb Z_p) =GL_2(\mathbb Q_p)/GL_2(\mathbb Z_p)$ as the set of vertices of a tree (the Bruhat-Tits tree) of degree $p+1$.
The group $\Gamma_p$ acts on this tree ...
- 148k
6
votes
Accepted
Is there a notion of hyperbolicity for number rings?
My intuition is that there will not be a precise definition of a hyperbolic number field. However, there there may be some number fields you can confidently say are hyperbolic.
Consider the following ...
- 148k
6
votes
Accepted
Subgroups of $Sp(2n,\mathbb{R})$ between $Sp(2n,\mathbb{Z})$ and some arithmetic group
First of all, thank you for the adjective "fantastic"(!). The question is actually studied in a paper of mine (link to the MR review). Sorry for talking about my own paper (I have no option, since I ...
- 11.2k
6
votes
Accepted
Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$
I think the reference to Millson-Raghunathan may be a little misleading. All that is needed is that if $\Gamma \subset G(\mathbb Z)$ is an arithmetic group in a linear algebraic group $G$ defined over ...
- 11.2k
6
votes
Accepted
Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
(Essentially copied from comments)
It's just Borel-Harish-Chandra + Borel's density theorem (+ definition of Chevalley group). More precisely, $G$ is a $\mathbf{Q}$-defined group without nontrivial ...
- 63.9k
6
votes
Accepted
Adelization for any classical arithmetic subgroup
For a subgroup to have a meaningful lift to the adeles, it is necessary and sufficient for the subgroup to be a congruence subgroup in the sense that for some $N$, the subgroup contains all elements ...
- 148k
5
votes
Accepted
When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?
Why the condition $\ell = 1 \bmod N$ is necessary. Suppose $g \sigma_\ell g^{-1}$ lies in the double coset $\Gamma \sigma_\ell \Gamma$. Then $g \sigma_\ell g^{-1} = r \sigma_\ell s$ for some $r, s \in ...
- 37k
5
votes
Are finite presentations of arithmetic groups computable?
You might want to study the work of Detinko, Flannery, and co-authors. For example:
Detinko, A.; Flannery, D. L.; Hulpke, A., Zariski density and computing in arithmetic groups, ZBL06825254.
...
- 96.4k
5
votes
Accepted
Explicit construction of division algebras of degree 3 over $\mathbb{Q}$
Looking more carefully in Pierce - Associative algebras, I found the answer I was looking for, which I'm going to describe here for future reference.
The algebra $D$ in the question is a realisation ...
- 767
5
votes
Accepted
Multiplicative group of number field mod field norms of quadratic extension
The index is definitely not finite. If $F$ is an arbitrary number field and $E$ is a finite extension with $[E:F] > 1$, the norm subgroup ${\rm N}_{E/F}(E^\times)$ has infinite index in $F^\times$. ...
- 50.6k
5
votes
Accepted
What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
I computed the matrices for $n=3,...,10$ in Mathematica and found the factorizations of the characteristic polynomials:
$$-(1 + x) (1 - 18 x + x^2),$$
$$ 1 - 68 x - 122 x^2 - 68 x^3 +
x^4,$$ $$-(1 +...
- 68.8k
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