Tubular free by cyclic groups and the strongest Tits alternative

2015/10/20 by J. O. Button, Button, J. O.
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1510.05842

openalex publication_date 2015/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Some questions related to free-by-cyclic groups and tubular groups
• The Tits alternative for CAT(0) cubical complexes
• A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes
• Classifying Virtually Special Tubular Groups
• A note on abelian cubulated groups