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


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