This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" if $\Gamma$ is a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles in my previous question would yield a structure that can detect freeness of the underlying group.

This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" if $\Gamma$ is a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles in my previous question would yield a structure that can detect freeness of the underlying group.

This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" whether $\Gamma$ was a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles into my previous question will result in a structure that can detect freeness of the underlying group.

This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" if $\Gamma$ is a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles in my previous question would yield a structure that can detect freeness of the underlying group.