Distortion of distances in \(p\)-adic groups (Q1568127)

Let \(G\) be a compactly generated locally compact group and \(\Omega\) a compact symmetric neighbourhood of the identity which generates \(G\). Then \(\Omega\) defines a distance function on \(G\) by setting \(|g|_G =\inf\{n:g \in\Omega^n\}\) for \(g\in G\). Any two generating neighbourhoods give rise to equivalent distances. Now, let \(H\) be a closed compactly generated subgroup of \(G\). It is an interesting question of how distance functions on \(G\) and on \(H\) are related. This problem has been investigated for real Lie groups by \textit{N. Th. Varopoulos} [Distance distortion on Lie groups, Symposia Math. XXXIX, 320-348 (1999); C. R. Acad. Sci., Paris, Sér. I 322, 1025-1026 (1996; Zbl 0867.22006)]. In the paper under review the author studies connected algebraic groups \(G\) over the \(p\)-adic number field. First, suppose that \(H\) is a Zariski-closed compactly generated subgroup of \(G\). Then restrictions to \(H\) of distance functions on \(G\) and distance functions on \(H\) are equivalent (Théorème 1), that is, there exists \(c>0\) such that \({1\over c}|h|_G \leq|h|_H\leq c|h|_G\) for all \(h\in H\). Secondly, the same conclusion holds whenever \(H\) is a closed amenable subgroup of \(G\) such that both \(H\) and its Zariski-closure are compactly generated (Théorème 2). The proofs heavily exploit the structure theory of the groups involved. These results are in some contrast to those for real Lie groups. As a sample we mention one of Varopoulos's results. If \(H\) is a closed subgroup of a connected real Lie group \(G\), then the distortion between \(|\cdot|_G\) and \(|\cdot |_H\) can be arbitrarily large, whereas it is at most exponential (that is, \(|h|_H\leq c\exp (c|h|_G))\) whenever \(H\) is connected and amenable.