Profinite isomorphisms, stable commutator length, and fixed point properties

Abstract

We construct Grothendieck pairs witnessing that stable commutator length and quasimorphisms are not profinite invariants, answering a question of Echtler and Kammeyer. Moreover, we recover the fact that property FA, property NL (which obstructs actions on hyperbolic spaces), and property FW${}_{\infty }$ (which obstructs actions on finite-dimensional CAT(0) cube complexes) and the existence of non-abelian free subgroups are not profinite invariants. The method combines Rips constructions with iterated group-theoretic Dehn filling on hyperbolic virtually special groups.

arXiv comment22 pages. v2: small changes, in particular the profinite non-invariance of NL and FW_$\infty$ are now attributed to Cheetham-West--Lubotzky--Reid--Spitler