Profinite isomorphisms, stable commutator length, and fixed point properties

TL;DR

This paper constructs examples showing that several important group properties, including stable commutator length and fixed point properties, are not preserved under profinite completions, challenging previous assumptions in group theory.

Contribution

It introduces new Grothendieck pairs demonstrating that properties like stable commutator length and property FW$_ Infty$ are not profinite invariants, using advanced group-theoretic constructions.

Findings

Stable commutator length is not a profinite invariant.

Property FW$_ Infty$ is not a profinite invariant.

Non-abelian free subgroups are not profinite invariants.

Abstract

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