Introduction

In this paper four descriptions of one and the same quasi-isometry class of pseudo-metrics on a reductive group G over a local field are given. They are as follows. The first one is the word metric corresponding to a compact set of generators of G. The second one is the pseudo-metric given by the action of G by isometries on a metric space. That these two pseudo-metrics on a group G are quasi-isometric holds in great generality. The third pseudo-metric is defined using the operator norm for a representation ρ of G. This pseudo-metric depends very much on the representation. But for a reductive group over a local field it does not up to quasi-isometry. The fourth pseudo-metric is given on a split torus over a local field K by valuations of the K*–factors. The main result is that these four pseudo-metrics on a reductive group over a local field coincide up to quasi-isometry. We thus have four different descriptions of one and the same very natural and distinguished quasi-isometry class of pseudo-metrics.

This paper contains foundational material for joint work in progress with G. A. Margulis on the following two topics. One is work on the following question of C. L. Siegel's. Given a reductive group G over a local field and an S–arithmetic subgroup Γ of G, it is one of the main results of reduction theory to describe a fundamental domain R for Γ in G, a so called Siegel domain.