Monthly Archives: May 2011

A finitely presented non-hyperbolic subgroup of a hyperbolic group

It is natural to inquire whether hyperbolicity is inherited by subgroups. This question is easily seen to have a negative answer; finitely generated free groups of rank at least two are hyperbolic but have subgroups, such as their commutator subgroups, … Continue reading

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Dehn functions of subgroups of CAT(0) groups

This is a post based on a guest lecture by Pallavi Dani. Introduction In this post, we will discuss Dehn functions of subgroups of non-positively curved groups. Let be a finitely presented group and let be a presentation complex for … Continue reading

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Fibre products and the membership problem

Baumslag, Bridson, Miller and Short (BBMS) provide criteria for what they call the fibre products to be finitely presented. They then leverage these results to show that, amongst other things, there exists a torsion-free hyperbolic group with a fibre product … Continue reading

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Distortion of finitely presented subgroups of non-positively curved groups II — hydra groups

This post is written by Tim Riley. (Given that I’m writing about my own work here, it would seem awkward to blog this under the partial anonymity of the collective name “Berstein.”) The main sources for this post are my … Continue reading

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Distortion of finitely presented subgroups of non-positively curved groups I

We have seen that the Rips construction can be used to construct hyperbolic groups with finitely generated subgroups of enormous distortion — that is, not bounded above by any recursive function. However, those examples were not finitely presentable. This post … Continue reading

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The Tits alternative and non-positive curvature

The following is largely based on the paper The Tits Alternative for CAT(0) Cubical Complexes by Sageev and Wise. 1.1. The Tits alternative A group satisfies the Tits alternative if for all subgroups , either is virtually solvable or it … Continue reading

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