All Questions
Tagged with geometric-group-theory coxeter-groups
18 questions
3
votes
0
answers
308
views
Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
Richard Schwartz considered ...
- 2,083
12
votes
0
answers
479
views
Writing an element of a free product of $C_2$'s as a product of order-$2$ elements
My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
- 1,298
8
votes
1
answer
486
views
When are groups generated by reflections in a triangle discrete?
Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
- 68.9k
3
votes
0
answers
117
views
Why inherit the Tits systems structure by a $B$-adapted homomorphism?
Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
- 12.9k
4
votes
0
answers
182
views
Polynomial invariants of infinite reflection groups
It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
- 41
10
votes
3
answers
666
views
Subgroups of RAAGs vs. subgroups of RACGs
Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group?
It is well-known from the theory of special cube complexes that ...
- 8,401
4
votes
1
answer
205
views
Subgroups of $W(E_8)$
Are there any proper subgroups of the Coxeter group $W(E_8)$ which are also proper overgroups of $W(A_8)$, other than $\text{Aut}(A_8)$?
- 12.9k
6
votes
3
answers
506
views
Order from Coxeter-Dynkin diagram
How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
- 1,846
6
votes
0
answers
99
views
Indices of Coxeter groups in themselves
Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...
- 2,773
9
votes
1
answer
493
views
Can $E_8$ be enlarged?
Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
- 3,587
0
votes
0
answers
185
views
Isomorphic Coxeter groups
After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
- 2,773
6
votes
1
answer
189
views
Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups
Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...
- 8,493
1
vote
0
answers
130
views
Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
- 63.9k
2
votes
1
answer
222
views
Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$
Looking at the images below, you recognize that the adjacency matrix of the graph $A_G$ splits up into three different colored submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...).
It's ...
- 14.5k
2
votes
1
answer
192
views
Relation between Riemannian and Cayley-graph distance in a finite Coxeter group
Background: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial ...
CommunityBot
- 1
4
votes
1
answer
398
views
Torsion-free, normal subgroups of certain Coxeter groups
Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...
- 63.9k
7
votes
1
answer
526
views
Generators of pure braid groups of arbitrary Coxeter groups
Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...
- 534
2
votes
0
answers
211
views
A question of braid words
Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...
- 534