Decompositions of locally compact contraction groups, series and extensions

A locally compact contraction group is a pair (G, alpha), where G is a locally compact group and alpha: G -> G an automorphism such that alpha(n)(x) -> e pointwise as n -> 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, alpha) which are central extensions {0} -> F-p((t)) -> G -> F-p((t)) -> {0} of the additive group of the field of formal Laurent series over F-p = 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. (C) 2020 Elsevier Inc. All rights reserved.