Atlas: Classification of the simple totally disconnected contraction groups by Helge Glockner

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21st Summer Conference on Topology and its Applications
July 6-9, 2006
Georgia Southern University
Statesboro, GA, USA

Organizers
Martha Abell, Francis Jordan, Frédéric Mynard, Sze-Man Ngai

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Classification of the simple totally disconnected contraction groups
by
Helge Glöckner
Darmstadt
Coauthors: George A. Willis (Newcastle, N.S.W.)

If G is a locally compact topological group and a: G→ G a contractive automorphism (i.e., aⁿ(x)→ 1 as n→∞, for each x ∈ G), then (G, a) is called a contraction group. It is known that each contraction group is a direct product G=G₀×D of its connected component G₀ and an a-stable closed subgroup D which is totally disconnected [2]. Therefore the study of contraction groups splits into the cases of connected groups (fully discussed in [2]) and totally disconnected groups. In the talk, I'll present results concerning the structure of totally disconnected contraction groups, obtained jointly with G. A. Willis [1]:

1. Any totally disconnected contraction group admits a composition series

1 = G₀ < G₁ < ... < G_n = G

of a-stable closed subgroups, and the composition factors are determined up to topological isomorphism. Each composition factor is a simple contraction group, i.e., it does not have proper, non-trivial, a-stable closed normal subgroups.

2. The main result is a classification of the simple, totally disconnected contraction groups. Besides p-adic vector spaces (with linear automorphisms), there are only countably many, parametrized by the finite simple groups.

3. As a consequence of the classification, each totally disconnected contraction group is a direct product G=tor(G)×div(G), where tor(G) is the subgroup of torsion elements and div(G) the subgroup of infinitely divisible elements. The torsion group tor(G) has finite exponent, and div(G)=G_p₁×...×G_{p_n} for certain a-stable unipotent p-adic algebraic groups G_p.

[1] Glöckner, H. and G. A. Willis, Classification of the simple factors appearing in composition series of totally disconnected contraction groups, arXiv:math.GR/0604062.

[2] Siebert, E., Contractive automorphisms on locally compact groups, Math. Z. 191 (1986), 73-90.

Darmstadt University of Technology, Department of Mathematics,
Schlossgartenstr. 7, 64289 Darmstadt, Germany

Date received: April 21, 2006