We say $G$ is the *Zappa-Szep product* of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective.

The Iwasawa decomposition shows that we can have amenable $K$ and $P$ such that $G=KP$ is non-amenable. However in such examples $K$ is usually not amenable when considered as a discrete group.

Where is a good place for me to look for examples of "naturally occurring" Zappa-Szep products where the two "factors" are amenable as discrete groups, yet the larger group itself is non-amenable? Can this ever happen for e.g. lattices in semisimple Lie groups?