A Haagerup Inequality for $\tA_1\times\tA_1$ and $\tA_2$ Buildings

TL;DR

This paper establishes Haagerup-type inequalities for groups acting on $ A_1 imes A_1$ and $ A_2$ buildings, providing new examples of higher rank groups with property (RD).

Contribution

It proves the first inequalities of this kind for higher rank groups acting on these specific buildings, extending Haagerup's inequality beyond free groups.

Findings

Derived inequalities for groups on $ A_1 imes A_1$ and $ A_2$ buildings.

First examples of higher rank groups with property (RD).

Applicable to groups of automorphisms acting transitively on vertices.

Abstract

Haagerup's inequality for convolvers on free groups may be interpreted as a result on $\tA_1$ buildings, i.e. trees. Here are proved analogous inequalities for discrete groups acting freely on the vertices of $\tA_1\times\tA_1$ and $\tA_2$ 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).