Cloning systems and action operads

Abstract

Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.

v3: Accepted version, to appear in Publicacions Matemà tiques. Minor changes and some additions to Subsections 3.2 and 3.3. v2: In this version we have removed an assumption on the definition of cloning system, matching the original definition. Additionally, we have expanded the section on action operads