Operad groups and their finiteness properties

TL;DR

This paper introduces a unifying operad-based framework for Thompson-like groups, demonstrating that certain operad groups with finiteness conditions are of type $F_$, thus extending previous results in the literature.

Contribution

It develops a category-theoretic framework for operad groups and proves they are of type $F_$ under specific finiteness conditions, unifying prior results.

Findings

Operad groups with finiteness conditions are of type $F_$.

The framework unifies various Thompson-like groups.

Extends existing proofs of finiteness properties.

Abstract

We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.