All Questions
Tagged with free-groups geometric-group-theory
44
questions
2
votes
1
answer
28
views
Transition matrix associated to representative of element in $Out(F_n)$
There is a notion of transition matrix associated to elements in $Out(F_n)$ from Bestvina and Handel's paper that I am a little bit confused.
Let $\Phi\in Out(F_n)$ and $\phi:\Gamma\to\Gamma$ a ...
- 689
2
votes
1
answer
33
views
Every graph morphism that is an immersion and surjective on the fundamental group is a homeomorphism
Here, we consider graphs as 1-dimensional CW complex and a graph morphism is a map sending vertices to vertices and $[f(a),f(b)]=f([a,b])$ where $[a,b]$ represents an edge connecting vertices $a,b$. A ...
- 689
4
votes
0
answers
67
views
$SO_3(\mathbb Q)$ contains a free group using 5-adic numbers
I am trying to show that $SO_3(\mathbb Q)$ contains a free group using 5-adic numbers, and more precisely using the matrices $M_1=\left(\begin{array}{ccc}1 &0&0\\0&\frac 35&-\frac{4}5\\...
- 41
2
votes
1
answer
49
views
If a graph map is an immersion, then the induced homomorphism on fundamental groups is injective
So I was reading some Geometric group theory and came across Stalling's folding of graphs. Now I am trying to use the folding idea to prove that every finitely generated subgroup of a free group is ...
- 377
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
4
votes
0
answers
182
views
A Conjecture in Low-Dimensional Topology.
Context
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
2
votes
0
answers
97
views
$F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$.
In a paper I read that $F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$. The proof of this fact is as follows:
Choose $F_2\leq \mathrm{Aut}(F_2)$ and let it act diagonally on $F_2^{2n-4}$, ...
- 1,932
0
votes
0
answers
30
views
Checking the image of mapping class in $\text{Aut}(F_{2g})$ stabilizes boundary curve
Overview: the mapping class group maps into $\text{Aut}(F_{2g})$ and its image stabilizes the surface relation. I am trying to check this for a specific example and am doing something wrong.
The ...
- 326
5
votes
1
answer
78
views
Epimorphism between free groups that inject on a finite subset
I asked a question on MathOverflow (https://mathoverflow.net/q/454012/513011) where the following lemma appeared:
Folklore lemma: Let $S$ be a finite subset of the free group $F_n$ of finite rank $n$....
- 167
4
votes
0
answers
47
views
Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(\mathbb F_d,\...
- 15.5k
1
vote
0
answers
93
views
Questions to limit groups (over free groups)
My questions refer to the following article (both refer to page 27):
https://arxiv.org/pdf/2002.10278.pdf
In the article we find the statement that for a non-abelian limit group $L$ we always find a ...
- 167
3
votes
1
answer
155
views
Definition of hyperbolic elements and axes in a (limit) group
I am writing on my master thesis at the moment and it is based on the following article of Fujiwara and Sela: https://arxiv.org/abs/2002.10278
On page 7 they use the terms "hyperbolic element&...
- 167
3
votes
0
answers
88
views
Proof of Uniqueness of Free Group [closed]
I just wanted to verify if my proof is correct as I have to present this proof in my next class.
Consider the comm. diagram drawn above. We can verify that it commutes due to the categorical ...
0
votes
0
answers
65
views
Conditionally negative definite functions on free groups
Let $G$ be a group. I call a function $\ell\colon G\to [0,\infty)$ a conditionally negative definite (cnd) length function if
$\ell(g)=0$ iff $g=e$,
$\ell(g)=\ell(g^{-1})$,
$\sum_{g,h\in G}\overline{\...
- 15.5k
0
votes
0
answers
56
views
Order of generators of commutator subgroup
Let $H$ be commutator subgroup of a free group $G$. This free group $G$ is generated by $a$ and $b$ such that both have infinite order. We know that $H$ is not finitely generated. However, can we ...
- 297