On realizing HNN groups in 3-manifolds

Realizing arithmetic invariants of hyperbolic 3–manifolds

These are notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the last of three “Volume Conferences,” held at Columbia University, LSU Baton Rouge, and Columbia University respectively in March 2006, May/June 2007, June 2009. The first part of the lecture series was expository, and since most of the material is readily ava...

Ascending Hnn - Extensions and Properly 3 - Realisable Groups

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 = ...

On Groups which contain no HNN-Extensions

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 23 We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 25 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine...

Subgroups of Quasi-hnn Groups

Hyperbolic Structures on 3-manifolds, Ii: Surface Groups and 3-manifolds Which Fiber over the Circle

The main result (0.1) of this paper is that every atoroidal three-manifold that fibers over the circle has a hyperbolic structure. Consequently, every fibered three-manifold admits a geometric decomposition. The main tool for constructing hyperbolic structures on fibered three-manifolds is the double limit theorem (4.1), which is of interest for its own sake and lays out general conditions unde...