All Questions
Tagged with free-groups at.algebraic-topology
8 questions
9
votes
2
answers
939
views
A question on the fundamental group of a compact orientable surface of genus >1
Let $G=\pi(X,x)$ be the fundamental group of a compact orientable
surface of genus $g\ge 2$. It is well known that a presentation of
$G$ is
$$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots
[x_g,...
- 1,386
6
votes
0
answers
128
views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
- 52.6k
5
votes
1
answer
264
views
Bases of surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
- 11.3k
4
votes
2
answers
337
views
A Karrass-Solitar theorem for surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...
- 11.3k
3
votes
1
answer
445
views
Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)
What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...
- 504
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 $H \leq \Gamma_g$ ...
- 11.3k
2
votes
1
answer
226
views
Lefschetz numbers for homomorphisms of free groups
Let $G = F_X$ be the free group on a finite set $X$, and $\phi\colon G\to G$ a group homomorphism. Consider the number
$$ \sum_{x\in X} (\text{number of occurrences of the generator $x$ in the word $...
- 66.8k
1
vote
0
answers
275
views
Explicitly showing that a free group is LERF [closed]
Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...
- 11.3k