Questions in Geometric Group Theory — Q 12.20

Definitions and mathematical statement

Let $F_n$ be the rank-$n$ free group, with $n\geq3$, and let $G=\operatorname{Out}(F_n)$ with a word metric from a finite generating set. A quasi-isometry $f:G\to G$ distorts distances by a multiplicative constant and an additive constant and has coarsely dense image. Quotient these maps by the relation $\sup_{x\in G}d(f(x),f'(x))<\infty$ to obtain the group $QI(G)$ under composition.

Left translations define a natural homomorphism \[ G\longrightarrow QI(G),\qquad g\longmapsto [x\mapsto gx]. \] Is this an isomorphism? Thus the question asks both whether translations give distinct classes and whether every self quasi-isometry is a bounded distance from a left translation.

The associated geometric question asks whether every self quasi-isometry coarsely preserves maximal flats: subsets given by cosets of maximal-rank free abelian subgroups. For each such subset $A$, must its image have finite Hausdorff distance from another such subset $A'$? Finite Hausdorff distance means each set is contained in a bounded neighborhood of the other.

Coarse preservation is the relevant interpretation: arbitrary bounded perturbations of a map cannot be expected to send these discrete subsets exactly onto one another. The rank restriction excludes the exceptional low-rank outer automorphism groups. These questions concern all self quasi-isometries, rather than only automorphisms or commensurations already defined algebraically.

Short English statement

Are all large-scale symmetries of the outer automorphism group induced by group elements, and do they preserve maximal abelian flats?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101220 (AMR-010-1220), 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 12.20 (PDF page 23). Original source indexed by AMR-010-1220. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.