When are graphs of cohomologically complete groups cohomologically complete?

2013-01-20T18:43:57Z

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 $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow H^\ast(G, \mathbb{Z}_p).$

In the paper "3-manifold groups are virtually residually-p", http://arxiv.org/abs/1004.3619 Aschenbrenner and Friedl define a $p$-efficient graph of groups $G$ to have the following properties:

1) $\pi_1(G)$ is residually-$p$ finite.

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 $G$;

3) for all vertices v and edges e, the pro-p topology on $G$ induces the pro-$p$ topology on $G_v$ and on $G_e$.

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.

On the other hand, an earlier paper "Groups with the same cohomology as their pro-$p$ completions" http://arxiv.org/pdf/0809.3046v3.pdf by Lorensen said the following for amalgamated products and HNN extensions:

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.)

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)?

Thanks very much for your time,

Kevin

When are graphs of cohomologically complete groups cohomologically complete? - MathOverflow

When are graphs of cohomologically complete groups cohomologically complete?