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?