Questions in Geometric Group Theory — Q 11.4

Definitions and mathematical statement

For an integer $n\geq1$, let $F_n$ be the free group on $x_1,\ldots,x_n$, and let $\phi\in\operatorname{Aut}(F_n)$. Its mapping-torus group is the semidirect product \[ M_\phi=F_n\rtimes_\phi\mathbb Z =\langle x_1,\ldots,x_n,t\mid tx_it^{-1}=\phi(x_i),\ 1\leq i\leq n\rangle. \] This algebraic definition is independent, up to the evident convention on $t$, of a chosen graph map representing the automorphism.

Equip each finitely generated group with a word metric. A map $f:X\to Y$ is a quasi-isometry if there are constants $L\geq1,C\geq0$ such that \[ L^{-1}d_X(x,x')-C\leq d_Y(fx,fx')\leq Ld_X(x,x')+C \] for all $x,x'$, and every point of $Y$ is within distance $C$ of $f(X)$. The quasi-isometry type of a finitely generated group is independent of its finite generating set.

Classify all groups $M_\phi$ up to quasi-isometry, allowing both the automorphism and finite rank to vary. A complete criterion must decide when two such groups have the same large-scale metric type. It need not identify the stable letters or the distinguished free kernels, and the quasi-isometry need not be a homomorphism. The question covers polynomially growing and exponentially growing automorphisms alike; restricting to hyperbolic mapping tori leaves part of the classification unaddressed.

Short English statement

Classify mapping-torus groups of free-group automorphisms by their quasi-isometry types.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101104 (AMR-010-1104), 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 11.4 (PDF page 19). Original source indexed by AMR-010-1104. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.