A Haagerup Inequality for ${\text{tA}}_1\times {\text{tA}}_1$ and ${\text{tA}}_2$ Buildings

Haagerup’s inequality for convolvers on free groups may be interpreted as a result on ${\text{tA}}_1$ buildings, i.e. trees. Here are proved analogous inequalities for discrete groups acting freely on the vertices of ${\text{tA}}_1\times {\text{tA}}_1$ and ${\text{tA}}_2$ buildings. The results apply in particular to groups of type-rotating automorphisms acting simply transitively on the vertices of such buildings. These results provide the first examples of higher rank groups with property (RD).