We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.

We show that the objects of Bass-Farrell categories which represent 0 in the corresponding Nil groups are precisely those which are stably triangular. This extends to Waldhausen's Nil group of the amalgamated free product with index 2 factors. Applications include a description of Cappell's special UNil group and reformulations of those splitting and fibering theorems which use the Nil groups.

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