Questions tagged [free-groups]
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8 questions from the last 365 days
16
votes
2
answers
602
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$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
2
votes
1
answer
137
views
Indicability of $\mathrm{Out}(F_n)$
A group $G$ is said to be indicable if it surjects onto $\mathbb{Z}$.
If $n=1$: $\mathrm{Out}(F_1)=\mathbb{Z}/2\mathbb{Z}$ and no finite group surjects onto an infinite group.
If $n\geq 4$: $\...
- 911
4
votes
1
answer
189
views
Equation in the conjugacy class of a free group
I will pose the question in the form in which it originally appeared to me:
Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...
- 111
2
votes
0
answers
115
views
Test words in free profinite groups
Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
- 355
4
votes
1
answer
305
views
Extending primitive systems in free groups
It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \...
- 41
9
votes
1
answer
526
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Shortest almost trivial element of free group [duplicate]
Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$.
Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.
What is the ...
- 45k
10
votes
1
answer
555
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Can automorphism equivalence in a free group be detected in a nilpotent quotient?
If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...
- 9,722
13
votes
1
answer
548
views
Generators for the first cohomology of free groups
Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...
- 195