Consider the following closure/containment properties for a class of groups $\mathcal{G}$.
- If $G, H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$.
- If $G \in \mathcal{G}$, $G$ is f.p., and $H$ is r.p. and quasi-isometric to $G$, then $H \in \mathcal{G}$.
Here, f.g. means finitely generated, f.p. means finitely presented, r.p. means recursively presented.
What interesting/surprising/non-trivial corollaries do closure properties 1 and 2 have?
The motivation for this question is that I have a class for which I believe I can prove 1 and 2, so I'd like to know what interesting things my class contains just based on that. I already know a very interesting corollary, namely that my class contains a simple group (Burger-Mozes). Unbelievable! Are there other surprises?
If it simplifies things, don't get too hung up on the r.p. and f.p. details. For instance if you can say something about the title, i.e. QI-closure of NA$\times$NA.