All Questions
Tagged with free-groups reference-request
27
questions
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81
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Has anyone studied differential equations on $\mathbb{Z}F_n$ defined using Fox derivatives?
I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied ...
- 2,382
4
votes
0
answers
47
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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
5
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answers
128
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Is there a sense in which $(\mathbb{R},+)$ is "free"?
Given a set $S$, the free group on $S$ consists of finite strings of elements in $S$. They can be visualized as paths on the integer grid $\mathbb{Z}^S$ starting at the origin, with the group ...
user519413
1
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answers
423
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Representations of free groups
What is known about the representation theory of free groups? In particular, I am interested in character theory and faithful representations. Many results in this theory seem to depend on the group ...
- 861
5
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1
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134
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Good introduction to free groups and free products
In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products.
Right now, ...
- 2,663
2
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82
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References on Free Profinite Groups
I would like to study free profinite groups (i.e., profinite completion of free groups) and I am looking for some introductory materials on this topic. Can anyone give me some recommendations?
- 5,451
2
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2
answers
416
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Schreier transversal and a basis for commutator subgroup of $F_3$
I've seen the calculation for a Schreier transversal and basis for $[F_2,F_2]\lhd F_2=\langle x,y\rangle$, but these groups aren't so complex that the calculations were particularly illuminating. I ...
- 288
3
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1
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103
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Reference for an exercise in free groups
It is a simple exercise to prove that if $F$ is the free group on two letters $\{x,y\}$, then the subgroup $$H = \{w \in F \mid \text{sum of all }y\text{ exponents in } w = 0 \} = \text{ normal ...
- 2,965
3
votes
1
answer
144
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automorphism of free group $F_n$ fixing $F_{n-1}$ and image of $n$th generator
I am reading a proof of the following statement:
Let $F_n$ be the free group of rank $n$, generated by a basis $\{x_1, \ldots, x_n\}$ and $\Phi$ an automorphism of $F_n$. Let $F_{n-1}$ be the ...
- 4,488
4
votes
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87
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An algorithm determining whether two subgroups of a free group are automorphic
In the book Lyndon, Schupp, Combinatorial Group Theory, the edition from 2000 P.30 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two subgroups ...
- 701
3
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1
answer
114
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Recovering a word in the free group of rank two from its image
The second paragraph of Zubkov (1998) [1] reads (see the free preview here):
As usual, by a Sanov representation we mean a homomorphism of the free group $F(x, y)$ with the free generators $x$ and ...
- 1,058
11
votes
1
answer
463
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Asymptotic length of reduced word on free group with replacements
This seems to be an elementary question, but it's proving hard for me to just Google. Suppose you have a sequence which picks elements out of $\{a, a^{-1}, b, b^{-1}, c, c^{-1}\}$ with equal ...
- 1,058
0
votes
0
answers
55
views
Reference needed for limiting word of random walk on free group
I'm looking for a reference on random walks on free groups that gives some kind of theorem about existence of a limiting word. A specific case I'm interested in is the free group on two letter $a,b$, ...
- 28.1k
3
votes
1
answer
228
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when two homomorphism on a free group is conjugate
Let $F_2=\langle a, b\rangle$ be the non-abelian free group on two generators. Let $\phi_1, \phi_2: F_2\to F_2$ be two group homormophisms. It may be vague, but I still want to ask the following ...
- 3,693
0
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1
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58
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Reference for a theorem of Serre
I recently came to know a theorem of Serre, which I couldn't search online with proof.
If $G$ is torsion-free and contains a free subgroup of finite index, then $G$ is free.
Can one provide a ...
- 10.3k