# Questions tagged [arithmetic-groups]

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### Abelianization of $\mathrm{GL}_n(\mathbb{Z})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$What is the abelianization of $\GL_n(\mathbb{Z})$? I know the abelianzation of $\GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field and the ...
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### Abelianizations of arithmetic Fuchsian groups

Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l &...
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### Variants of Selberg trace formula

I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...
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### Stabilizer in $G(\mathbb{Z})$ of point in fundamental domain $G(\mathbb{Z}) \backslash G(\mathbb{R}) / K$

Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(\mathbb{R})$, and ...
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### Conway big picture for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$

I saw in Conway’s paper "Understanding groups like $\Gamma_0(N)$" that the so-called Big Picture can give simple interpretations for important objects in number theory, such as Hecke ...
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### Subgroups of Sp(2g,Z) that map onto all Sp(2g,Z/m)

I stumbled into the following problem. I apologize for being a bit naive. For $g\geq 3$, consider the group $\mathrm{Sp}(2g,\mathbb{Z})$ of symplectic square matrices of order $2g$ with integral ...
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### Colimits in cohomology of profinite arithmetic groups

Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...
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1 vote
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### Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?

I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices. Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
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### Euler characteristic with compact support of spaces of Euclidean lattices

Has the Euler characteristic with compact support of $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ been computed ? References? Thanks.
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### Is this a lattice?

Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...
1 vote
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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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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  $\Gamma$ ...
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}...