Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H\_i(\Gamma;\mathbb{Z})=0$ for all $i \in S$.


The only positive answer I know here is that $S=\{1\}$ works since every countable group can be embedded into a simple group. I don't know any negative answers.


I'm especially interested in singleton sets $S$ (in particular, $S=\{2\}$ and $S=\{3\}$).


Also, is the question easier if I restrict myself to finitely generated or finitely presentable groups?



Embedding groups into groups with some vanishing homology groups
See Baumslag, G.; Dyer, E.; Miller, C. F.
On the integral homology of finitely presented groups. 
Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 321–324, and the full version Baumslag, G.; Dyer, E.; Miller, C. F., III On the integral homology of finitely presented groups. Topology 22 (1983), no. 1, 27–46. Lemma 4 in particular.


