Decompositions of locally compact contraction groups, series and extensions

TL;DR

This paper studies the structure and classification of locally compact contraction groups, showing how they can be decomposed, extended, and constructed using cohomology and 2-cocycles, revealing a rich diversity of such groups.

Contribution

It proves the existence of continuous sections for homomorphisms, describes extensions via cohomology, and constructs many non-isomorphic examples using 2-cocycles.

Findings

Every surjective equivariant homomorphism admits a continuous global section.

Extensions with abelian kernels can be described by continuous equivariant cohomology.

Uncountably many non-isomorphic totally disconnected contraction groups are constructed using 2-cocycles.

Abstract

A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends to infinity. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,f) which are central extensions of the additive group of the field of formal Laurent series over Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.