Questions in Geometric Group Theory — Q 7.5

Definitions and mathematical statement

A group is torsion-free if every nonidentity element has infinite order, and finitely presented if it has a presentation with finitely many generators and relations. A nontrivial one-edge splitting of $G$ over $C$ is a reduced amalgamated product $A*_C B$ or an HNN extension with associated subgroup $C$, with the edge inclusions injective. Equivalently it is a minimal action on a simplicial tree, without inversions or a global fixed point, with one orbit of edges and an edge stabilizer isomorphic to $C$.

Is there a torsion-free finitely presented group $G$ which admits no such splitting over any virtually abelian subgroup, but which admits infinitely many distinct splittings over the rank-two free group \[ F_2=\langle a,b\mid\ \rangle? \] A virtually abelian group has an abelian subgroup of finite index; the prohibition includes cyclic and trivial edge groups. Distinct splittings are understood through their Bass--Serre trees, up to $G$-equivariant simplicial isomorphism after suppressing inessential subdivisions. Merely changing the names of generators in a displayed presentation does not create a new splitting.

The same ambient group must supply the infinite collection. No bound is imposed on the ranks or presentations of its vertex groups. This asks whether the absence of low-complexity abelian splittings can coexist with infinitely many splittings over one fixed nonabelian free edge-group type.

Short English statement

Can a finitely presented torsion-free group have infinitely many rank-two free splittings while having no virtually abelian splitting?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100705 (AMR-010-0705), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 7.5 (PDF page 16). Original source indexed by AMR-010-0705. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.