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.

The Bianchi groups are a family of discrete subgroups of PSL2(C) which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.

An extraordinary theorem of Gromov, [4], characterizes the finitely generated groups of polynomial growth; a group has polynomial growth iff it is nilpotent by finite. This theorem went a long way from its roots in the class of discrete subgroups of solvable Lie groups. Wolf, [11], proved that a polycyclic group of polynomial growth is nilpotent by finite. This theorem is primarily about linear...

The most useful constructions in Combinatorial Group Theory are amalgamated free products and HNN-extensions, and they are the two basic examples in the theory of graphs of groups due to Bass and Serre (see [9]). We recall that given a group G and a subgroup H 6 G together with monomorphisms (respectively homomorphisms) ψ, φ : H −→ G, the group determined by the presentation 〈 G, t; t−1ψ(h)t = ...

We develop a version of Bass-Serre theory for Lie algebras (over field k) via homological approach. define the notion fundamental algebra graph and show that this construction yields Mayer-Vietoris sequences. extend some well known results in group to N-graded algebras: example, we one relator are iterated HNN extensions with free bases which can be used cohomology computations apply sequence g...