The Karoubi-Weibel complexity for groups

TL;DR

This paper introduces the Karoubi-Weibel complexity for finitely presented groups, inspired by topological methods, and explores its properties, calculations, and applications to geometric group theory problems.

Contribution

It defines a new combinatorial complexity for groups, extending topological concepts to group theory, and demonstrates its usefulness in geometric applications.

Findings

Defined the Karoubi-Weibel complexity for groups.

Calculated or estimated the complexity for specific group classes.

Applied the complexity to problems in systolic area and volume entropy.

Abstract

Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, 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.