[Paper Review] Automorphisms and homology of non-positively curved cube complexes

Corey Bregman|arXiv (Cornell University)|Sep 12, 2016

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper introduces a new invariant called the 'genus' for special cube complexes, which characterizes when the fundamental group is abelian (genus one) or surjects onto F₂. Using this invariant, the authors provide a new geometric proof that special groups are either abelian or surject onto a non-cyclic free group, and further show that the Torelli subgroup of a right-angled Artin group is torsion-free via geometric automorphism actions on homology.

ABSTRACT

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.

Motivation & Objective

• To define and study a new geometric invariant, the genus, for special cube complexes to understand the structure of their fundamental groups.
• To provide a new geometric proof of the result that special groups are either abelian or surject onto F₂, resolving a question posed by Wise.
• To investigate automorphisms of special cube complexes and their action on first homology, particularly focusing on the Torelli subgroup.
• To establish that the Torelli subgroup of a right-angled Artin group is torsion-free using geometric realization of automorphisms and homology non-triviality.

Proposed method

• Define the genus of a special cube complex as the maximal number of pairwise disjoint, non-separating hyperplanes whose union does not disconnect the complex.
• Prove that a special cube complex has genus one if and only if its fundamental group is abelian, linking the geometric invariant to algebraic structure.
• Use the genus invariant to show that any non-abelian special group surjects onto F₂, providing a new geometric proof of a result originally established algebraically.
• Construct a blow-up of the standard Salvetti complex to realize automorphisms of special groups as cube complex automorphisms.
• Analyze the action of automorphisms on first homology by showing that any automorphism preserving all hyperplanes must be trivial, hence non-trivial actions on homology imply non-trivial automorphisms.
• Apply the geometric realization of automorphisms to prove that any finite-order outer automorphism acting trivially on homology must be trivial, hence the Torelli subgroup is torsion-free.

Experimental results

Research questions

• RQ1 What is the geometric meaning of the genus invariant in special cube complexes, and how does it relate to the algebraic structure of the fundamental group?
• RQ2 Can the genus invariant be used to give a new geometric proof that special groups are either abelian or surject onto F₂?
• RQ3 Which automorphisms of a special group can be realized as automorphisms of a cube complex, and when do they act non-trivially on first homology?
• RQ4 Is the Torelli subgroup of a right-angled Artin group torsion-free, and can this be shown using geometric actions on cube complexes?

Key findings

• The genus of a special cube complex is one if and only if its fundamental group is abelian, providing a geometric characterization of abelian special groups.
• A special cube complex has genus zero if and only if it is CAT(0), linking the genus to non-positive curvature.
• The genus of the fundamental group of a closed surface of genus g is equal to g, showing consistency with the classical topological genus.
• The genus gives a lower bound on the corank of the group, and the paper raises the open question of whether genus equals corank in general.
• Every automorphism of a blow-up of a Salvetti complex acts non-trivially on first homology, which is key to proving the torsion-freeness of the Torelli subgroup.
• The Torelli subgroup of a right-angled Artin group is torsion-free, as any finite-order outer automorphism acting trivially on homology must be trivial, proven via geometric realization of automorphisms on cube complexes.