All Questions
Tagged with free-groups covering-spaces
21
questions
2
votes
0
answers
35
views
$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point
Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
- 430
0
votes
0
answers
44
views
Covering space of compact surface with free fundamental group
Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?
I know that if we ...
1
vote
0
answers
66
views
How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?
In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
- 545
7
votes
2
answers
299
views
The fundamental group of closed orientable surface of genus 2 contains a free group on two generators
Let $S$ be the closed orientable surface of genus $2$. It is well known that its fundamental group is given by $$ \pi_1(S)=\langle a,b,c,d:[a,b][c,d]=1\rangle.$$
How can we show that this group has a ...
- 1,630
1
vote
0
answers
98
views
Rank of subgroups of free groups
I am familiar with Schreier's theorem which states that any subgroup H of a free group G is free, as well as with the formula
$$
\operatorname{rank} H - 1 = |G:H|(\operatorname{rank} G - 1)
$$
I have ...
- 75
0
votes
1
answer
42
views
Minimal generating set of $p_*(\pi_1(E,e))$
Consider the following degree $4$ non-normal covering space of $S_1\lor S^1$ I drew:
Here, $a$ and $b$ denote the edges which map onto the first and second circle in $S^1\lor S^1$ respectively. I ...
- 598
5
votes
1
answer
135
views
Given a subgroup of a free group, find the associated covering space.
Let $R_2$ the rose with $2$ petals, that is the wedge of $S^1$ with itself. We know its fundamental group is the free group with two elements, $\pi_1(R_2)=F_2=\langle a,b\rangle$. Now given some ...
1
vote
0
answers
71
views
What is the induced homomorphism of the covering map $p : Y \to X : z \mapsto z^3 - 3z$ on the fundamental groups?
Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map
$$
p : Y \to X : z \mapsto z^3 - 3z
$$
is a 3-branched covering as given in this question of Math ...
- 11.9k
12
votes
2
answers
151
views
Can a space with $\pi_1(X)=F_2$ have a nontrivial covering space which is homeomorphic to $X$?
Any finite sheeted connected cover of the circle is again homeomorphic to a circle. On the group level, this is consistent with the fact that subgroups of $\mathbb Z$ are isomorphic to $\mathbb Z$. ...
- 27.5k
1
vote
1
answer
71
views
Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$
$\newcommand{\Z}{\mathbb Z}$
Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
- 2,454
3
votes
1
answer
292
views
Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$
Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$.
This is a follow-up to my post from yesterday regarding the kernel K of a map $\Phi: ...
- 2,454
4
votes
1
answer
240
views
Torus is the only closed orientable surface regularly covered by punctured plane
Let $\Sigma_g$ be the closed orientable surface of genus $g$. There is
no covering map $p\colon\Bbb R^2\backslash \mathbf 0\to \Sigma_g$ so that $p_*\pi_1(\Bbb R^2\backslash \mathbf 0)$ is a normal ...
- 649
3
votes
1
answer
659
views
Normal covering spaces of the wedge sum of $n$ circles
Exercise 1.31 in Hatcher's Algebraic Topology states the following:
Show that the normal covering spaces of $S^1 \vee S^1$ are precisely the graphs that are Cayley graphs of groups with two generators....
- 1,503
1
vote
1
answer
339
views
Covering space of Bouquet of Two circles
Given a bouquet of 2 circles $B_2$, is there a way to find a covering space of $B_2$ such that the deck transformation group of our covering space is isomorphic $F_2 / <R>$. i.e. the group $G = \...
- 687
4
votes
1
answer
573
views
finite index subgroups in free group non-trivial intersection with each of the non-trivial subgroups of the free group.
I was reading a paper and find this statement in the abstract, "If $H$ has finite index in $F_m$, then $H$ has non-trivial intersection with each of the non-trivial subgroups of $F_m$" where $F_m$ is ...
- 1,090