All Questions
Tagged with geometric-group-theory nt.number-theory
8 questions
12
votes
2
answers
1k
views
Group generated by two irrational plane rotations
What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$?
If the centers of the rotations coincide, then the rotations commute and generate some ...
- 1,277
2
votes
1
answer
224
views
Parahoric subgroup over a local field
$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...
- 479
4
votes
0
answers
288
views
Commutator algorithm
Let $M \in \mathrm{SL}(2, \mathbb{Z}).$ Is there an efficient algorithm to write $M$ as a commutator (group commutator, not algebra commutator) [or fail if this is impossible]?
Addendum: answering ...
- 96.4k
5
votes
2
answers
407
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
4
votes
2
answers
190
views
Orbits of some special actions on solution set of a arithmetic equation
Let $g_1(x,y,z)=(y,x,-z), g_2(x,y,z)=(y,x+y+2z,-y-z)$,
$V= \{(x,y,z)\in Z^3|xy-z^2+1=0 \}$.
Is it possible to find all orbits of the action of group $\langle g_1 \rangle * \langle g_2 \rangle$ on $V$? ...
- 191
13
votes
1
answer
910
views
Holomorphic cusp forms and cohomology of GL(2,Z)
Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
- 40.2k
8
votes
3
answers
2k
views
Links between Geometric Group Theory and Number Theory
Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.
7
votes
2
answers
639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
