# Central extensions of torsion groups by $\mathbb{R}^n$

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.

• Can't you write $\Gamma$ as direct limit of finite groups and check if $H^2(\lim\limits_{\rightarrow}\Gamma_i,\mathbb{R})=\lim\limits_{\leftarrow}H^2(\Gamma_i,\mathbb{R})=0$?
– tj_
Sep 22, 2021 at 1:10
• Not every torsion group is a direct limit of finite groups. Perhaps $\Gamma$ is supposed to be locally finite? Sep 22, 2021 at 1:16
• @MarkSapir I don't mean to say $\Gamma$ is locally finite, I think this is why the search for a counterexample has been difficult. If every element of $\Gamma$ is contained in a finite subgroup then $H^2(\Gamma,\mathbb{R}^n) = 0$. This is due to the fact that $\alpha$ will vanish when restricted to any finite subgroup. Sep 22, 2021 at 7:06
• The canonical splitting you describe is characterized by the property that the associated 2-cocycle $\Gamma\times\Gamma\to\mathbf{R}^n$ vanishes on $H\times H$ for every finite cyclic subgroup $H$ of $\Gamma$, and satisfies the property that it vanishes on $H\times H$ for every finite subgroup $H$ of $\Gamma$.
– YCor
Sep 22, 2021 at 7:11
• In my previous comment, I mean to say "every pair of elements of" $\Gamma$ not "every element". Every element of $\Gamma$ is contained in a finite cyclic subgroup, of course. Sep 22, 2021 at 7:21

The paper S. I. Adyan and V. S. Atabekyan, V. S. Central extensions of free periodic groups, Mat. Sb. 209 (2018), no. 12, 3–16; translation in Sb. Math. 209 (2018), no. 12, 1677–1689 proves that if $$n\geq 665$$ is odd and $$m\geq 2$$, then the Schur multiplier $$H^2(B(m,n),\mathbb Z)$$ for the free Burnside group $$B(m,n)$$ of exponent $$n$$ on $$m$$-generators is free abelian of countable rank. In the arXiv version this is Corollary 3 and so using the universal coefficients theorem, $$H^2(B(m,n),\mathbb R^n)$$ is quite huge. If you want an explicit class, then that is beyond my competency range.