  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.
