Embedding groups into groups with some vanishing homology groups - MathOverflow

Embedding groups into groups with some vanishing homology groups

2011-04-22T04:40:06Z

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?

Answer by Mariano Suárez-Alvarez for Embedding groups into groups with some vanishing homology groups

2011-04-22T04:48:12Z

Every countable group can be embedded in a countable algebraically closed group, and the latter is acyclic.

It follows that all subsets of $\mathbb N$ have the property you want :)

Answer by Mark Sapir for Embedding groups into groups with some vanishing homology groups

2011-04-24T07:53:23Z

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.

Answer by Chris Gerig for Embedding groups into groups with some vanishing homology groups

2011-04-24T10:17:42Z

To add to Mark Sapir's post, the answer is precisely given as Corollary 5.6 of $\textit{The Topology of Discrete Groups}$ by Baumslag, Dyer, Heller (JPAA 16, 1980):

"Every countable group can be embedded in a 7-generator acyclic group."

Thus all possible $S$ work.