All Questions
10 questions
9
votes
1
answer
212
views
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 ...
- 17.4k
7
votes
1
answer
397
views
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 $\Lambda$ ...
user1688
7
votes
1
answer
550
views
Explicit construction of division algebras of degree 3 over $\mathbb{Q}$
In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/...
- 767
5
votes
2
answers
409
views
Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$
I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:
Specifically, ...
- 6,025
5
votes
0
answers
267
views
Generating congruence subgroups of SL_n over totally imaginary number rings
Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...
- 51
4
votes
1
answer
274
views
Subgroups of $Sp(2n,\mathbb{R})$ between $Sp(2n,\mathbb{Z})$ and some arithmetic group
The fantastic answers to my previous question Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup led me to the following question.
Let $O_K$ be the ring of ...
- 195
2
votes
2
answers
293
views
Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$
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.
...
- 391
1
vote
2
answers
312
views
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 ...
- 1,277
1
vote
0
answers
67
views
Cardinality or covolume of $S$-units in quaternion algebras
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 ...
- 391
1
vote
0
answers
95
views
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 $\Gamma$ ...
- 1,101