Let $Gamma$ be a torsion group (i.e. ever 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 counter example probably exists.