2
votes
1answer
67 views

Show the existence of a certain subgroup of F2

This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
2
votes
1answer
91 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
2
votes
0answers
36 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
2
votes
2answers
135 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
2
votes
1answer
63 views

Double cover of symplectic groups

What is the normal definition of double cover of Symplectic group? I couldn't find a simple and understandable definition
2
votes
0answers
56 views

is a free cellular action of a discrete group over a cellular complex always properly discontinuous?

I think that the answer is "yes" for free simplicial actions over simplicial complexes: a non trivial element g of the group G must map a simplicial simplex to a different one, because of brower ...
10
votes
2answers
925 views

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
3
votes
0answers
225 views

Double Coverings of the Double Torus

I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is $$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$ where ...
5
votes
2answers
262 views

The only finite group which can act freely on even dimensional spheres is $C_2$.

I don't know how to show this. Do I assume $G$ acts on $S^{2n}$ by homeomorphisms? Then, since $S^{2n}$ is Hausdorff I'd know $G$ acts freely and properly discontinuously, and since ...
2
votes
0answers
52 views

Dehn Twist in the sense of graphs

Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or subgroups. I want to know ...