  This paper deals with a class of totally disconnected groups acting on
buildings, among which are certain Kac-Moody groups. The apartments of our
buildings are hyperbolic planes tiled by right-angled polygons. We discuss
linearity properties for the groups, as well as an analogy with semisimple
groups over local fields of positive characteristic. Looking for
counter-examples to this analogy leads to the construction of Moufang twinnings
"with several ground fields".
