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# Questions tagged [arithmetic-groups]

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### Inheritance of arithmeticity properties in orbifold strata

Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
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### Lattices in $p$-adic groups

What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank? It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a ...
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### Multiplicative group of number field mod field norms of quadratic extension

I'm reading some notes(*) about arithmetic lattices in $\operatorname{SU}(n,1)$. I'm trying to understand the data that classifies (up to commensurability) the arithmetic lattices of the "first ...
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### Clarification on arithmetic groups example

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$I'm working through some of the constructions in Introduction to Arithmetic Groups by Dave Witte Morris, and I'm confused by the construction of ...
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### Index of subgroups of $\mathrm{Sp}(4,\mathbb{Z})$ conjugate in $\mathrm{GL}(4,\mathbb{Q})$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($...
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### What is known about the cohomology of the U-duality group?

$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
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### The principal congruence subgroup of the symplectic group over the integers

Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ... 225 views

### Covolumes of unit groups of division algebras

Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group ...
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### Adelization for any classical arithmetic subgroup

In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups. I have ...
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### 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 ...
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### 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 ...
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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 $\mathrm{SL}_{2}(\mathcal{... 1 vote 0 answers 173 views ### Matrix factorizations over$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}$admits several factorizations (the polar decomposition, the KAN decomposition, the Schur triangular form, etc). Those ... 4 votes 1 answer 249 views ### Unitary representations of lattices Let$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$is a finitely generated ... 5 votes 1 answer 281 views ### Free abelian subgroups of$\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})$? It cannot be bigger that the virtual cohomological dimension of$\...
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 ...