1
vote
0answers
63 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
2
votes
2answers
105 views

Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
17
votes
1answer
307 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
0
votes
1answer
42 views

The deficiency of surface group

Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The ...
1
vote
1answer
164 views

Is there an example of a non-orientable group manifold?

Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
2
votes
1answer
114 views

Could a surface bundle over a circle have free fundamental group?

Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this ...
3
votes
0answers
109 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
1
vote
0answers
61 views

Fundamental Domain of Manifold Reference Request

I am interested in learning about the fundamental domain of a manifold and I am wondering if anyone know of any papers or descriptions online other than Wikipedia and the linked articles? I am looking ...
7
votes
3answers
371 views

Putting Geometries on Knot Complements

I have two different, but related, questions about the type of geometry one can get on a knot complement. Quickly some notation: $K$ will be a non-trivial smooth knot - living in $S^3$ - and $M$ will ...