Contraction groups and scales of automorphisms of totally disconnected locally compact groups

Abstract

We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.

Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces

“…This lemma is a consequence of the study of the so called tidy subgroups [2]. Lemma 2 (See [3], [17]) Assume G ⊂ GL(d, F) is compactly generated, F is non-archimedean, and any g ∈ G generates a bounded subgroup.…”

Classification of the simple factors appearing in composition series of totally disconnected contraction groups

“…A contraction group is a pair (G, α), where G is a topological group and α : G → G a contractive automorphism, meaning that α n (x) → 1 as n → ∞, for each x ∈ G. Contraction groups arise in probability theory on locally compact groups (see, e.g., [10]), representation theory ( [14], [15], [16], [21]), and the structure theory of locally compact groups initiated in [22] (see [1] and [7]). It is known from the work of Siebert that every locally compact contraction group is a direct product G = G e × D of a connected group G e and an α-stable totally disconnected group D, whence the study of locally compact contraction groups splits into the two extreme cases of connected groups and totally disconnected groups (see [20,Proposition 4.2]).…”

Contraction Groups for Tidy Automorphisms of Totally Disconnected Groups

“…Contraction groups arise in the study of semistable convolution semigroups on second countable groups [6], [8]. Contraction groups in p-adic Lie groups have been investigated in [2], and recently general results for metrizable totally disconnected groups G were obtained, using the concept of a tidy subgroup (see [10], [11]) as a tool [1]. We recall the definition.…”