All Questions
Tagged with geometric-group-theory at.algebraic-topology
5 questions
24
votes
1
answer
968
views
Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
20
votes
4
answers
2k
views
Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
6
votes
1
answer
406
views
Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
3
votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...