Groups with the same cohomology as their pro-$p$ completions

Details Groups with the same cohomology as their pro-$p$ completions Groups with the same cohomology as their pro-$p$ completions

2008-09-18

arxiv.org/abs/0809.3046v3

J. Pure Appl. Algebra 214 (2010), 6-14

Mathematics

Group Theory

The revisions in the second version pertain to the exposition: the proof of Prop. 1.1, in particular, now includes more detail

Scientific paper

For any prime $p$ and group $G$, denote the pro-$p$ completion of $G$ by $\hat{G}^p$. Let $\mathcal{C}$ be the class of all groups $G$ such that, for each natural number $n$ and prime number $p$, $H^n(\hat{G^p},\mathbb Z/p)\cong H^n(G, \mathbb Z/p)$, where $\mathbb Z/p$ is viewed as a discrete, trivial $\hat{G}^p$-module. In this article we identify certain kinds of groups that lie in $\mathcal{C}$. In particular, we show that right-angled Artin groups are in $\mathcal{C}$ and that this class also contains some special types of free products with amalgamation.

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