Embedding groups into groups with some vanishing homology groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:34:53Z http://mathoverflow.net/feeds/question/62596 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62596/embedding-groups-into-groups-with-some-vanishing-homology-groups Embedding groups into groups with some vanishing homology groups Andy Putman 2011-04-22T04:40:06Z 2011-04-24T10:17:42Z <p>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 $i \in S$.</p> <p>The only positive answer I know here is that <code>$S=\{1\}$</code> works since every countable group can be embedded into a simple group. I don't know any negative answers.</p> <p>I'm especially interested in singleton sets $S$ (in particular, <code>$S=\{2\}$</code> and <code>$S=\{3\}$</code>).</p> <p>Also, is the question easier if I restrict myself to finitely generated or finitely presentable groups?</p> http://mathoverflow.net/questions/62596/embedding-groups-into-groups-with-some-vanishing-homology-groups/62598#62598 Answer by Mariano Suárez-Alvarez for Embedding groups into groups with some vanishing homology groups Mariano Suárez-Alvarez 2011-04-22T04:48:12Z 2011-04-22T04:48:12Z <p>Every countable group can be embedded in a countable algebraically closed group, and the latter is acyclic.</p> <p>It follows that <em>all</em> subsets of $\mathbb N$ have the property you want :)</p> http://mathoverflow.net/questions/62596/embedding-groups-into-groups-with-some-vanishing-homology-groups/62814#62814 Answer by Mark Sapir for Embedding groups into groups with some vanishing homology groups Mark Sapir 2011-04-24T07:53:23Z 2011-04-24T08:03:51Z <p>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.</p> http://mathoverflow.net/questions/62596/embedding-groups-into-groups-with-some-vanishing-homology-groups/62828#62828 Answer by Chris Gerig for Embedding groups into groups with some vanishing homology groups Chris Gerig 2011-04-24T10:17:42Z 2011-04-24T10:17:42Z <p>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):</p> <p>"Every countable group can be embedded in a 7-generator acyclic group."</p> <p>Thus all possible $S$ work.</p>