When are graphs of cohomologically complete groups cohomologically complete? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:05:23Z http://mathoverflow.net/feeds/question/119418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119418/when-are-graphs-of-cohomologically-complete-groups-cohomologically-complete When are graphs of cohomologically complete groups cohomologically complete? kevinschreve 2013-01-20T18:43:57Z 2013-01-21T15:22:33Z <p>A group $G$ is cohomologically $p$- complete if the canonical map from $G$ to it's pro$-p$ completion $\hat G^p$ induces an isomorphism on cohomology $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow H^\ast(G, \mathbb{Z}_p).$</p> <p>In the paper "3-manifold groups are virtually residually-p", <a href="http://arxiv.org/abs/1004.3619" rel="nofollow">http://arxiv.org/abs/1004.3619</a> Aschenbrenner and Friedl define a $p$-efficient graph of groups $G$ to have the following properties:</p> <p>1) $\pi_1(G)$ is residually-$p$ finite.</p> <p>2) for all vertices v and edges e, the subgroups $G_v$ and $G_e$ of G are closed in the pro-p topology on $G$;</p> <p>3) for all vertices v and edges e, the pro-p topology on $G$ induces the pro-$p$ topology on $G_v$ and on $G_e$.</p> <p>They then have the following lemma: Suppose $G$ is a $p$-efficient graph of finitely generated groups such that the vertex and edge groups are cohomologically $p$-complete. Then $\pi_1(G)$ is cohomologically $p$-complete.</p> <p>On the other hand, an earlier paper "Groups with the same cohomology as their pro-$p$ completions" <a href="http://arxiv.org/pdf/0809.3046v3.pdf" rel="nofollow">http://arxiv.org/pdf/0809.3046v3.pdf</a> by Lorensen said the following for amalgamated products and HNN extensions: </p> <p>Let $G = G_1 \ast_{G_0} G_2$ and suppose $G_i$ is topologically p-embedded in G for each i, and $G_0$ is topologically p-embedded in $G_1$ and $G_2$. If $G_i$ are all cohomologically p-complete, then $G$ is cohomologically p-complete.(He has a similar statement for HNN extensions.)</p> <p>If I am reading this correctly, topologically p-embedded is exactly condition (3) above. So, my question is, are the other conditions necessary- particularly residually p-finiteness? Is there a difference between amalgamated products and general graphs of groups that requires conditions (1) and (2)?</p> <p>Thanks very much for your time, </p> <p>Kevin</p>