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A Cayley graph for F2 × F2 which is not minimally almost convex

    https://doi.org/10.1142/S0218196722500059Cited by:0 (Source: Crossref)

    We give an example of a Cayley graph Γ for the group F2×F2 which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for F2×F2 does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property K lying between FFTP and MAC (i.e., FFTPKMAC) is dependent on the generating set. This includes the well-known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru’s condition P(2) and the basepoint loop shortening property (LSP) for which dependence on the generating set was previously unknown. We also show that the Cayley graph Γ does not have the LSP, so this property also depends on the generating set.

    Communicated by O. Kharlampovich

    AMSC: 20F65

    References

    • 1. J. W. Cannon , Almost convex groups, Geom. Dedicata 22(2) (1987) 197–210. Crossref, Web of ScienceGoogle Scholar
    • 2. M. Elder , The loop shortening property and almost convexity, Geom. Dedicata 102(1) (2003) 1–17. Crossref, Web of ScienceGoogle Scholar
    • 3. M. Elder , Regular geodesic languages and the falsification by fellow traveler property, Algebraic Geom. Topol. 5 (2005) 129–134. Crossref, Web of ScienceGoogle Scholar
    • 4. M. Elder and S. Hermiller , Minimal almost convexity, J. Group Theory 8(2) (2005) 239–266. Crossref, Web of ScienceGoogle Scholar
    • 5. D. F. Holt and S. Rees , Shortlex automaticity and geodesic regularity in Artin groups, Groups Complex. Cryptology 5(1) (2013) 1–23. CrossrefGoogle Scholar
    • 6. I. Kapovich , A note on the Poénaru condition, J. Group Theory 5(1) (2002) 119–127. Web of ScienceGoogle Scholar
    • 7. R. C. Lyndon and P. E. Schupp , Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89 (Springer-Verlag, Berlin, 1977). CrossrefGoogle Scholar
    • 8. C. F. Miller and M. Shapiro , Solvable Baumslag–Solitar groups are not almost convex, Geom. Dedicata 72(2) (1998) 123–127. Crossref, Web of ScienceGoogle Scholar
    • 9. W. D. Neumann and M. Shapiro , Automatic structures, rational growth, and geometrically finite hyperbolic groups, Invent. Math. 120(1) (1995) 259–287. Crossref, Web of ScienceGoogle Scholar
    • 10. D. E. Otera and V. Poénaru, Topics in geometric group theory, Part I, preprint (2018), arXiv:1804.05216. Google Scholar
    • 11. V. Poénaru , Almost convex groups, Lipschitz combing, and π1 for universal covering spaces of closed 3-manifolds, J. Differential Geom. 35(1) (1992) 103–130. Crossref, Web of ScienceGoogle Scholar
    • 12. C. Thiel , Zur fast-Konvexität einiger Nilpotenter Gruppen, No. 234 (Mathematisches Institut der Universität Bonn, 1992). Google Scholar
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