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.

中文翻译：

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局部紧缩群、级数和扩展的分解

局部紧缩群是一对 (G,f)，其中 G 是局部紧群，f 是 G 的自同构，在 G 中每个 g 在 f 下的前向轨道收敛到中性元素 e 的意义上，它是收缩的，因为 n 趋于无穷大。我们表明，局部紧缩群之间的每个满射、连续、等变同态都允许等变连续全局部分。因此，具有阿贝尔核的局部紧缩群的扩展可以用连续等变上同调来描述。对于每个素数 p，我们使用 2-cocycles 来构造无数成对非同构的完全断开的局部紧缩群 (G,f)，它们是形式 Laurent 级数域在 Z/pZ 上的可加群的中心扩展通过它自己。相比之下，