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Profinite isomorphisms, stable commutator length, and fixed point properties

Francesco Fournier-Facio

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.

Profinite isomorphisms, stable commutator length, and fixed point properties

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 (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.
Paper Structure (11 sections, 14 theorems, 18 equations)

This paper contains 11 sections, 14 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group-theoretic Dehn filling
  4. Grothendieck pairs
  5. Norms
  6. A first construction
  7. Free subgroups and fixed point properties
  8. Fixed point properties
  9. The inductive step
  10. Proof of \ref{['algebraic:NL']}
  11. Concluding remarks

Key Result

Theorem 2.1

Let $G$ be a torsion-free hyperbolic group and let $P < G$ be an infinite, malnormal, quasiconvex subgroup. Then $G$ is hyperbolic relative to $P$.

Theorems & Definitions (32)

  • Theorem 2.1: bowditch
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4: gromovwise
  • Theorem 2.5: MSQT
  • Theorem 2.6: bridson
  • Theorem 2.7: macarena
  • Proposition 2.8
  • proof
  • Definition 2.9
  • ...and 22 more