[Paper Review] Tubular free by cyclic groups and the strongest Tits alternative

J. O. Button|arXiv (Cornell University)|Oct 20, 2015

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes that tubular free-by-cyclic groups act freely on CAT(0) cube complexes via Wise's equitable sets criterion and virtually satisfy the strongest Tits alternative—meaning every subgroup either surjects a non-abelian free group or is torsion-free abelian. The Gersten group is the first known example of such a group that is not virtually special nor virtually residually free, yet still virtually satisfies this strong group-theoretic property.

ABSTRACT

We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means that every subgroup either surjects a non abelian free group or is torsion free abelian. In particular the Gersten group is the first known group virtually having this property but which is not virtually special nor virtually residually free.

Motivation & Objective

• To determine whether tubular free-by-cyclic groups act freely on CAT(0) cube complexes.
• To investigate whether these groups satisfy the strongest Tits alternative, even when not virtually special or residually free.
• To identify the first example of a group that is not virtually special or virtually residually free but still virtually satisfies the strongest Tits alternative.
• To classify tubular groups that are free by cyclic using a homomorphism to ℤ non-zero on all edge groups.
• To explore the relationship between geometric actions on CAT(0) cube complexes and strong group-theoretic properties like the strongest Tits alternative.

Proposed method

• Use Wise’s equitable sets criterion to prove that tubular free-by-cyclic groups act freely on a CAT(0) cube complex.
• Construct a homomorphism to ℤ that is non-zero on all edge groups to characterize tubular groups that are free by cyclic.
• Apply the equitable sets condition to show that such groups satisfy the criterion for a free action on a CAT(0) cube complex.
• Prove that tubular groups with maximal edge inclusions virtually satisfy the strongest Tits alternative using structural group-theoretic analysis.
• Use finite covers to reduce to the case of maximal edge inclusions, enabling the application of Theorem 3.7.
• Leverage the fact that all free-by-cyclic tubular groups admit a finite cover with maximal edge inclusions to extend results to the general case.

Experimental results

Research questions

• RQ1 Do all tubular free-by-cyclic groups act freely on a CAT(0) cube complex?
• RQ2 Can the strongest Tits alternative be satisfied virtually by groups that are not virtually special or virtually residually free?
• RQ3 Is there a group that is not virtually special but still virtually satisfies the strongest Tits alternative?
• RQ4 What is the role of maximal edge inclusions in ensuring the strongest Tits alternative holds virtually?
• RQ5 How do geometric actions on CAT(0) cube complexes relate to strong group-theoretic properties like the strongest Tits alternative?

Key findings

• Every tubular free-by-cyclic group acts freely on a CAT(0) cube complex, as shown via Wise’s equitable sets criterion.
• The Gersten group is the first known example of a group that is not virtually special nor virtually residually free but still virtually satisfies the strongest Tits alternative.
• All tubular groups with maximal edge inclusions satisfy the strongest Tits alternative in a finite index subgroup.
• The finite index subgroup satisfying the strongest Tits alternative is explicitly constructed, with index 4 in the case of the Gersten group.
• The Woodhouse group and other examples are shown to have finite index subgroups satisfying the strongest Tits alternative, despite not acting freely on finite-dimensional CAT(0) cube complexes.
• The paper identifies that the strongest Tits alternative can be established without geometric cubulation, demonstrating that group-theoretic properties can be derived independently of geometric actions.