Abstract: Affine buildings are mostly useful because they admit nice actions by simple algebraic groups over local fields. However, there are some other constructions of 2-dimensional affine buildings, some of them with cocompact automorphism group.
After recalling the basic definitions and facts about affine buildings, I will explain the following theorem: a cocompact automorphism group of an $\tilde A_2$-building admits an infinite linear representation if and only if it is arithmetic. The proof uses ergodic techniques à la Margulis.
It is a joint work with Uri Bader and Pierre-Emmanuel Caprace.