Finiteness Properties of the Braided Thompson’s Groups and the Brin-Thompson Groups

A group G is of type Fn if there is a K(G, 1) complex that has finite n-skeleton. It is of type F∞, if it is of type Fn for all n ∈ N. Here the property F1 is equivalent to G being finitely generated and the property F2 equivalent to being finitely presented. An interesting question in the study of these finiteness properties is how they change, if the group under consideration is changed. One family of examples to consider, when attacking such a question, are Thompson’s groups, in particular F and V . It is well known, that both groups are of type F∞ and there are quite a few generalizations of Thompson’s groups in the literature. The question to consider here is whether these generalizations inherit the property of being of type F∞. In this thesis we will give an introduction to the classical Thompson’s groups F and V and discuss generalizations of them. In particular we will study the higherdimensional Brin-Thompson groups sV for s ∈ N and the braided Thompson’s groups Vbr and Fbr. We will prove that both generalizations inherit the property of being of type F∞. The proof of the Main Theorem requires the analysis of certain simplicial complexes. One family of complexes that we need to consider are generalizations of matching complexes of a graph to arcs on surfaces, that we introduce in this thesis. We will also give bounds on their connectivity properties for certain underlying graphs.