# Author Archives: berstein

## Farewell (and what did we miss?)

“I believe in everything until it’s disproved. So I believe in fairies, the myths, dragons.” — John Lennon Our tour of subgroups of non–positively curved groups has reached its end. What did we miss? Probably lots — apologies to any … Continue reading

## Cannon-Thurston maps for graphs of groups

Recall from our last post that given an infinite hyperbolic subgroup of a hyperbolic group , the extension of the inclusion to a continuous is called a Cannon-Thurston map. By the continuity of , if such a map exists it’s … Continue reading

## Boundaries of Hyperbolic Groups

In this post and the next, we will discuss Cannon-Thurston maps: (continuous) extensions of inclusions of hyperbolic groups to the boundaries . It is not evident that such maps should exist when is distorted in , but they often do … Continue reading

## Finitely presented subgroups of hyperbolic groups of cohomological dimension 2

Suppose is a group with group ring . A projective resolution of a –module is an exact sequence where each is a projective –module. Regard as a trivial –module. The cohomological dimension of a group is the smallest integer such … Continue reading

## 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

## 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

## 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