Let $\Gamma$ be a torsion group (i.e. every element has finite order). I am interested in understanding central extensions of the form: $\require{AMScd}$ \begin{CD} 0 @>>> \mathbb{R}^n @>\exp>> G @>\pi>> \Gamma @>>> 1\\ \end{CD} Equivalently, I want examples of groups $\Gamma$ with non-trivial classes in $H^2(\Gamma,\mathbb{R}^n)$. When $\Gamma$ is finite I'm aware that $H^2(\Gamma,\mathbb{R}^n) = 0$ but I have little intuition or examples for infinite torsion groups.

Interestingly, any central extension such a $\Gamma$ by $\mathbb{R}^n$ has a canonical set-theoretic splitting $\sigma \colon \Gamma \to G$ with the property that:

$$ \sigma(\gamma) = g \quad \Leftrightarrow \quad \pi(g) = \gamma \text{ and } g \text{ has finite order}$$ where $\pi \colon G \to \Gamma$ is the projection. It is not too hard to show that if any group-theoretic splitting exists then it must be $\sigma$.

Using usual group cohomology arguments, this gives rise to a cocycle: $$ \alpha \colon G \times G \to \mathbb{R}^n $$ I've managed to prove a few interesting properties of $\alpha$ (for example, $\alpha$ must be symmetric) but have not been able to show it is zero and I suspect a counterexample probably exists.

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