Waldhausen’s Nil groups and continuously controlled K-theory

Let ${\displaystyle \Gamma ={\Gamma }_1{\cdot }_G{\Gamma }_2}$ be the pushout of two groups ${\displaystyle {\Gamma }_i}$, i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces ${\displaystyle B{\Gamma }_1\leftarrow BG\leftarrow B{\Gamma }_2}$. Denote by ξ the diagram ${\displaystyle I\{pover\leftarrow \}H\{1over\to \}X=H}$, where p is the natural map onto the unit interval. We show that the ${\displaystyle Ni{l}^{\sim }}$ groups which occur in Waldhausen's description of ${\displaystyle K_{\cdot }(\mathbb{Z}\Gamma )}$ coincide with the continuously controlled groups ${\displaystyle w{\tilde{K}}^{cc}\_\cdot (\xi )}$, defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups ${\displaystyle w{\tilde{K}}^{cc}\_\cdot \left({\xi }^{+}\right)}$ which are known to form a homology theory in the variable ξ, with the "homology part" in Waldhausen's description of ${\displaystyle K_{\cdot -1}(\mathbb{Z}\Gamma )}$. A similar result is also obtained for HNN extensions.