Some remarks on Grothendieck pairs

Abstract

We revisit the paper of Alexander Grothendieck where he introduced Grothendieck pairs and discuss the relation between profinite rigidity and left/right Grothendieck rigidity. We also show that various groups are left and/or right Grothendieck rigid and, in particular, all ascending HNN-extensions of finitely generated free groups are right Grothendieck rigid. Along the way we present a number of questions and suggestions for further research.