Geodesic
On the geometry of the free factor graph for Aut$(F_{N})$
Mladen Bestvina1 orcid ; Martin R. Bridson2 orcid ; Richard D. Wade2 orcid
1University of Utah, Salt Lake City, USA
2University of Oxford, UK
Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 445-457

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DOI

Let Φ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface Σ with one boundary component. We show that if b∈π1​(Σ) is the boundary word, φ∈Aut(π1​(Σ)) is a representative of Φ fixing b, and adb​ denotes conjugation by b, then the orbits of 〈φ,adb​〉≅Z2 in the graph of free factors of π1​(Σ) are quasi-isometrically embedded. It follows that for N≥2 the free factor graph for Aut(FN​) is not hyperbolic, in contrast to the Out(FN​) case.
DOI : 10.4171/ggd/882
Classification : 20F65, 20E05, 20E36, 51F30
Mots-clés : free groups, automorphisms, Gromov-hyperbolic spaces, quasi-flats, group actions on trees

Mladen Bestvina  1   ; Martin R. Bridson  2   ; Richard D. Wade  2

1 University of Utah, Salt Lake City, USA
2 University of Oxford, UK 
Mladen Bestvina; Martin R. Bridson; Richard D. Wade. On the geometry of the free factor graph for Aut$(F_{N})$. Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 445-457. doi: 10.4171/ggd/882
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