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Let $G = \operatorname{SL}_2(\mathbb{Z})$. Consider an element $A \in G$ of trace $t$ of absolute value exceeding $2$, say. Let $C(A)$ denote the conjugacy class of $A$ in $G$. Let $c(A)$ denote the ...
Stanley Yao Xiao's user avatar
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In the group $G = \operatorname{SL}_2(\mathbb{Z})$, there are several basic operations, namely inverse, transpose, and conjugation. Certainly the whole group $G$ is closed under these operations, and ...
Stanley Yao Xiao's user avatar
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Question / conjecture Let $K$ be a real number field and consider a pair of real numbers $(x, x')$. Assume that there are real numbers $\epsilon > 0$, $C > 0$ and infinitely many pairs of real ...
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Source of considered construction: Morris - Introduction to Arithmetic Groups ; see (6.7.1). Let $F$ be a totally real algebraic number field and fix one real embedding $e: F \to \Bbb R$, such that $...
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In trying to understand a certain 5 term exact sequence, I formed three questions. What is $H_2(\operatorname{GL}_n(\mathbb{Z}); \mathbb{Q})$? This post's top answer claims that Dwyer and Mitchell ...
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Let $X$ be an irreducible affine variety over $\mathbb{Z}$. Suppose that $X(\mathbb{Z}) \subseteq X_{\mathbb{Q}}$ is Zariski dense. (UPD.: As suggested by the commentators, the last condition should ...
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$\DeclareMathOperator\SL{SL}\newcommand{\Z}{\mathbb{Z}}\newcommand{\bbF}{\mathbb{F}}$I'm trying to understand F. Calegari's paper here, and there is a technical point I find confusing. Here's the ...
Some random guy's user avatar
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Related question: Surfaces of general type with globally generated cotangent bundle. Computing the invariants of ball quotient surfaces. How far is ample from globally-generated. When is a general ...
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Is there a known classification of low-dimensional $\mathbb{Q}$-irreducible representations of $\operatorname{SL}(2,\mathbb{Z})$ over $\mathbb{Q}$ ? More specifically, following a discussion I had ...
Dima Pasechnik's user avatar
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$\DeclareMathOperator\Sh{Sh}$Notations Let $(G,X)$ be a Shimura datum where $G$ is define over $\mathbb{Q}$. Let $\mathbb{A}_{f}$ be the finite adeles over $\mathbb{Q}$. Let $\Sh_{K}(G,X)$ be the ...
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Let $p$ be a prime and consider the ring $R=\mathbb{H}(\mathbb{Z}[\frac{1}{p}])$ of quaternions over $\mathbb{Z}[\frac{1}{p}]$. I am interested in the group $G=R^{*}/\{p^n,n\in\mathbb{Z}\}$. The ...
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Is it generally known how to a compute a fundamental domain of a f.g. subgroup of $G:=\operatorname{SL}(n,\mathbb{Z})$, in the natural action on a space where $G$ itself has well-understood ...
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Fix $g\ge3$ and consider the Lie groups $$\mathrm{Sp}_{2g}(\mathbb{R})=\{A\in\mathrm{GL}_{2g}(\mathbb{R})\mid A^T\left(\begin{smallmatrix}0&\mathrm{Id}_g\\-\mathrm{Id}_g&0\end{smallmatrix}\...
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Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
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A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence ...
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Let $G$ be a semisimple Lie group of rank one and let $\Gamma$ be a convex-cocompact, Zariski dense subgroup. Let $X=G/K$ denote the symmetric space and $\partial X$ its visibility boundary. Let $\...
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$\DeclareMathOperator\SL{SL}$For the sake of this question, let's say that a group $G$ of finite cohomological dimension $n$ has a cohomological gap if for some $0 < i < n$ there is no subgroup $...
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I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
Noah B's user avatar
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$What is the abelianization of $\GL_n(\mathbb{Z})$? I know the abelianzation of $\GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field and the ...
Marcos's user avatar
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$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let $$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...
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Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane. Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it. Let $\Gamma=\Delta(p,q,...
zemora's user avatar
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Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant ...
Local's user avatar
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As $\rm{PSL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/3\mathbb{Z})$, its cohomology groups $H^n(\rm{PSL}(2,\mathbb{Z});\mathbb{Z})$ are easy to get. Let $N$ be a product of distinct primes. ...
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Let $D=\left(\frac{a,b}{\mathbb{Q}}\right)$ be a quaternion algebra over $\mathbb{Q}$. Suppose $D$ is ramified at a finite set $S$ of places and $\infty\in S$. It is known that the $S$-units (the unit ...
Jun Yang's user avatar
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I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
ಠ_ಠ's user avatar
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$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$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 ...
naf's user avatar
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Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
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$\DeclareMathOperator\SL{SL}$In Soulé's paper "The cohomology of $\SL_3(\mathbb{Z})$" the cohomology ring $H^*(\SL(3,\mathbb{Z}),\mathbb{Z})_{(2)}$ is determined in Theorem 4.iv. I'm wanting ...
Noah B's user avatar
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$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
Noah B's user avatar
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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
Noah B's user avatar
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I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
TSU's user avatar
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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'...
Ethan Dlugie's user avatar
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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 ...
Jun Yang's user avatar
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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$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^...
user267839's user avatar
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In Maclachlan-Reid we can read Let $G$ be a connected semisimple Lie group with trivial centre and no compact factor. Let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Then $\Gamma$ is ...
Jacques's user avatar
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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)...
Local's user avatar
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Let $\Lambda\subset \mathbb{R}^4$ be a lattice. We identify $\mathbb{R}^4$ with the space $M_2(\mathbb{R})$ of $2\times 2$ matrices over $\mathbb R$. It then is is clear that the set $$ \det(\Lambda)=\...
user avatar
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Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l &...
William of Baskerville's user avatar
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I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...
user482438's user avatar
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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups. Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
Andrew's user avatar
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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 ...
Ethan Dlugie's user avatar
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$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$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 ...
Ethan Dlugie's user avatar
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$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($...
Tee's user avatar
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$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
Arun Debray's user avatar
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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 ...
user avatar
8 votes
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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 ...
Radu T's user avatar
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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 ...
Radu T's user avatar
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$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
Gabriele Mondello's user avatar
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Let $K$ be a number field, which can be $\mathbb Q(\zeta)$ with $\zeta$ a root of unity if that helps, and let $\operatorname{SL}_n(\mathbb{Z})$ act on $(K^\times)^n:=K^\times\times\cdots\times K^\...
Mariano Suárez-Álvarez's user avatar
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If $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$, a subgroup of $\mathbf{G}(\mathbb{Q})$ is arithmetic if it is commensurable to $\mathbf{G}(\mathbb{Q}) \cap \operatorname{GL}_n(\mathbb{...
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