# Questions tagged [arithmetic-groups]

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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 $\Lambda$ ...
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### 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 ...
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### 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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### 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 ...
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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 ...
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. ...