Abstract

The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group Q and producing as an output a hyperbolic group G that maps onto Q with finitely generated kernel. The “output group” G is crafted by adding generators and relations to a presentation of Q, in such a way that these relations create enough “noise” in the presentation to ensure hyperbolicity. One can then lift pathological properties of Q to some subgroup of G. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem.

In this talk, I will explain Rips’ result, mention some of its variations, and survey some tools and concepts related to these constructions, including small cancellation theory, cubulated groups, and asphericity. I will also describe a new version of the Rips construction that produces cocompactly cubulated hyperbolic groups of any desired cohomological dimension.