The Karoubi–Weibel complexity for groups

Abstract

Let $G$ be a finitely presented group. A new complexity called Karoubi–Weibel complexity or covering type, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel (2016), initially applied to topological spaces. We introduce a similar notion in combinatorial form in order to apply it to finitely presentable groups. Some properties of this complexity as well as a few examples of calculation/estimation for certain classes of finitely presentable groups are considered. Finally, we give a few applications of complexity to some geometric problems, namely to the systolic area and the volume entropy of groups.