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The unitary representation theory of locally compact contraction groups andtheir semi-direct products with $\mathbb{Z}$ is studied. We put forward theproblem of completely characterising such groups which are type I or CCR andthis article provides a stepping stone towards a solution to this problem. Inparticular, we determine new examples of type I and non-type-I groups in thisclass, and we completely classify the irreducible unitary representations ofthe torsion-free groups, which are shown to be type I. When these groups aretotally disconnected, they admit a faithful action by automorphisms on aninfinite locally-finite regular tree; this work thus provides new examples ofautomorphism groups of regular trees with interesting representation theory,adding to recent work on this topic.

On the unitary representation theory of locally compact contraction groups