In the paper ["Hasse principle" for $PSL_2 (\mathbb Z)$ and $PSL_2(\mathbb F)$](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195506714) there's a definition of a **Hasse principle for a group $G$**, but I don't completely get it. Is there a more motivated reformulation of this definition?

Why I am interested: local-global principles are often very interesting in arithmetic geometry, so when I noticed a paper with this title I looked at it to see whether this proves something geometric. 

As said below, one can formulate a problem of computing a group $Sha$ defined by a $g$-module $G$ and a family of subgroups $h_i\in G$, but the actual computation in the paper is for a specific choice of $h_i$, and I can't parse if there is an application of interest. Is it so?