Selflessness, MIF and opposition in groups acting on exotic buildings

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type ${\tilde{A}}_2$ and ${\tilde{C}}_2$ are mixed-identity free and have selfless reduced $C^{\ast }$-algebras. These results follow from a strong form of ping-pong dynamics that we call ‘transversal contractivity’. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.