All Questions
Tagged with algebraic-groups arithmetic-groups
11 questions with no upvoted or accepted answers
10
votes
0
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465
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A uniform bound for a "true" non-congruence subgroup
Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
- 638
3
votes
0
answers
602
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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 ...
- 419
3
votes
0
answers
175
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$\text{PGL}_n(\mathbf{Q}_p)$ 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 $\...
user72293
2
votes
0
answers
80
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Question about lattice with dense projection
Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
- 305
2
votes
0
answers
111
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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 ...
- 332
1
vote
0
answers
151
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On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$
$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...
- 10.5k
1
vote
0
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98
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$K$-ranks of some algebraic groups in Lubotzky's "Discrete groups, expanding graphs and invariant measures"
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Let $G$ be a semisimple algebraic group and $K$ any field. Then
the $K$-rank of $G$ is the maximal rank $r$ of a $K$-splitting
torus $T \cong (K^...
- 5,966
1
vote
0
answers
227
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Why are compact arithmetic surfaces defined through quaternion algebras (usually) only over $\mathbb{Q}$?
As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)...
- 128
1
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0
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257
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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 ...
- 885
1
vote
0
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190
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Compactifications of group schemes
Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
- 41
0
votes
0
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121
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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}...
- 255