All Questions
Tagged with free-groups graph-theory
16
questions
2
votes
1
answer
28
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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
2
votes
1
answer
49
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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
0
votes
1
answer
90
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Lee Mosher book definition of a tree.
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?...
user965463
0
votes
0
answers
72
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Banach-Tarski Paradox: Extension with cycles
I am new to StackExchange so apologies if my question is poorly asked or does not abide by the standards.
Referring to the 2016 edition of Tomkowicz and Wagons' book on the Banach-Tarski Paradox, ...
6
votes
0
answers
106
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Question on the standard algebraic topology proof that a subgroup of a free group is free [duplicate]
I have a question on the standard algebraic topology proof that a subgroup of a free group is free. My understanding of that proof (mostly from Hatcher's topology) is as follows:
We define a topology ...
- 2,068
0
votes
1
answer
94
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If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.
If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.
Definition. A path-connected space whose fundamental group is isomorphic to a given group $G$ and which has ...
- 2,454
1
vote
1
answer
151
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Proving Neilsen-Schreier Theorem using "only free groups act freely on a tree"
This is exercise II.9.16 from Aluffi's Algebra: Chapter 0.
Before tackling this theorem, I have proved (rather loosely because I don't know much about graph theory) that the Cayley diagram of a free ...
- 131
1
vote
0
answers
647
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Free Groups and Actions on Trees
Here is the proof of a theorem I am working through in Geometric Group Theory by Clara Loh:
The first paragraph shows that if $F$ is free, then it admits a free action on a (non-empty) tree. This ...
- 8,040
3
votes
1
answer
112
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components of subgraph determine subgroups up to conjugacy
I am reading "The Tits alternative for $\operatorname{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms" by Bestvina, Feighn and Handel. In this paper, the following is written:
Suppose ...
- 4,488
2
votes
1
answer
431
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Bass-Serre tree of Isom($\mathbb{Z}$)
I wish to draw the Bass-Serre tree associated to Isom($\mathbb{Z}$). Now I'll be honest, I'm not absolutely certain what a Bass-Serre tree is, but this is in the context of amalgamated free products, ...
- 363
4
votes
1
answer
442
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Group Presentations and Cayley graphs
I am trying to understand group presentations and Cayley graphs, and have a few questions I am confused about.
Let $G=(V,E)$ be a finite $d$-regular graph that is known to be a Cayley graph for the ...
- 2,517
-1
votes
2
answers
679
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The graph of free product group. [closed]
For the free product $A*B$=G, where $A$ and $B$ are groups, there is a graph defined by: the edge set E(G)$\backsimeq$G and the vertex set V(G)$\backsimeq$G $G/A \bigsqcup G/B$, and to g=$a_1b_1......
- 3,726
3
votes
3
answers
836
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Is every normal subgroup of a finitely generated free group a normal closure of a finite set?
Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$.
My question is, if $G$ is a free ...
- 5,965
1
vote
3
answers
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Cayley graph is a tree iff group is free
I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S.
For the direction '$\implies $' I see that they have assumed that there are two ...
- 1,118