# Questions tagged [free-groups]

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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 ...
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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 ...
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### 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 ...
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### What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?

Follow up question, edited in on 12/20 below: Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
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### Almost free group without the Specker group as a subgroup

An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...
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### Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
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### A generalisation of residual finiteness?

A group $\Gamma$ is Residually Finite (RF) if $\forall g \neq e \in \Gamma$ there is a homomorphism $h: \Gamma \to G$ where $G$ is a finite group such that $h(g) \neq e$. Free groups are known to be ...
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