A cubical Rips construction

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• Macarena Arenas, University of Cambridge
• Friday 11 February 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Macarena Arenas.

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. Time permitting, I will describe a variation of the Rips construction that produces cubulated hyperbolic groups of any desired cohomological dimension.

This talk is part of the Junior Geometry Seminar series.

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