A Haagerup Inequality for $ \tilde A_1 \times \tilde A_1 $ and $ \tilde A_2 $ Buildings (original) (raw)

A Haagerup Inequality for $ \tilde A_1 \times \tilde A_1 $ and $ \tilde A_2 $ Buildings

Haagerup's inequality for convolvers on free groups may be interpreted as a result on $ \tilde A_1 $ buildings, i.e. trees. Here are proved analogous inequalities for discrete groups acting freely on the vertices of $ \tilde A_1 \times \tilde A_1 $ and $ \tilde A_2 $ buildings. The results apply in particular to groups of typerotating automorphisms acting simply transitively on the vertices of such buildings. These results provide the first examples of higher rank groups with property (RD).