On contraction groups of automorphisms of totally disconnected locally compact groups

Abstract

We prove that recent results of Baumgartner and Willis on contraction groups of automorphisms of metrizable totally disconnected locally compact groups (Israel J. Math. 142 (2004), 221-248) remain true for nonmetrizable groups.Let G be a (Hausdorff) topological group and τ an automorphism of G. The subgroupis called the contraction subgroup of τ . Recent work of Baumgartner and Willis [1] demonstrates the significance of the contractions subgroups in the theory of totally disconnected locally compact groups, by linking these subgroups with the theory of tidy subgroups and scales of automorphisms [1, Section 3]. This link is established under the extra assumption that the group be metrizable. It is the purpose of this note to prove that the results of Baumgartner and Willis remain true, without any modifications, for any totally disconnected locally compact group.A careful scrutiny of the proofs in [1] reveals that it is enough to prove that Theorem 3.8 of [1] remains true for not necessarily metrizable groups. With this accomplished, the proofs of the remaining results in [1, Section 3] do not require any changes.

Contraction groups and passage to subgroups and quotients for endomorphisms of totally disconnected locally compact groups

“…If α(M ) = M , we say that M is α-stable. This terminology is in line with [1] but differs from [13]. A sequence (x n ) n∈N0 (or two-sided sequence (x n ) n∈Z ) in a topological space X is called bounded if {x n : n ∈ N 0 } (resp., {x n : n ∈ Z}) is relatively compact in X.…”

The Concentration Function Problem for Locally Compact Groups Revisited: Nondissipating Space-Time Random Walks, -Decomposable Laws, and Their Continuous Time Analogues

“…And we have ( ) = ( ) ⋅ , at least in the case of totally disconnected groups or of Lie groups. See [11]; see also [13][14][15] for previous results. For Lie groups see [5], [6, Theorem 3.2.13].…”

“…For the history of the concentration function problem for random walks on locally compact groups the reader is referred to the survey of Jaworski [1] showing previous developments and a recent state of investigations: beginning with the pioneer works [16][17][18][19][20] to the investigations [1,11,13].…”

Endomorphisms of Lie Groups over Local Fields

“…Choose a ∈ ]0, 1[ such that a > |λ| K for all λ. Then g = g <a with respect to L(α For α an automorphism, the existence of small tidy subgroups is equivalent to closedness of −→ con (α) (see [1,Theorem 3.32] for the case of metrizable groups; the general case can be deduced with arguments from [37]). The following result concerning endomorphisms is sufficient for our Lie theoretic applications.…”

“…An isomorphism between totally disconnected contraction groups (G, α) and (H, β) is a continuous group homomorphism φ : G → H such that β • φ = φ • α. A totally disconnected contraction group (G, α) is called simple if G = {e} and G does not have closed α-stable normal subgroups other than {e} and G. Remark 9.2 We mention that contraction groups −→ con (α) of automorphisms arise in many contexts: In representation theory in connection with the Mautner phenomenon (see [40, Chapter II, Lemma 3.2] and (for the p-adic case) [55]); in probability theory on groups (see [28], [51], [52] and (for the p-adic case) [10]); and in the structure theory of totally disconnected, locally compact groups (see [1], [37], and [9]).…”