We say $G$ is the [*Zappa-Szep product*](https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_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?



Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?
Here is an observation of Yair Glasner that I previously missed.


Let $k$ be the field of real algebraic numbers and consider $G=\text{PSL}\_2(k)$.
Let $P=B(\mathcal{k})$ be the standard Borel subgroup and $K=\text{PSO}\_2(k)$.
Then $K$ and $P$ are solvable, $K\cap P=\{e\}$ and $G=KP$, but $G$ is not amenable (as it contains free groups).
All countable examples I am aware of are variants of this example.
In particular, I am not aware of any finitely generated example.


