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Journal Articles International Journal of Algebra and Computation Year : 2022

A Cayley graph for F 2 × F 2 which is not minimally almost convex

Andrew Elvey Price
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Abstract

We give an example of a Cayley graph Γ for the group F 2 × F 2 which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for F 2 × F 2 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., FFTP ⇒ K ⇒ MAC) 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 for which dependence on the generating set was previously unknown. We also show that the Cayley graph Γ does not have the loop shortening property, so this property also depends on the generating set.
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Dates and versions

hal-03886498 , version 1 (06-12-2022)

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Andrew Elvey Price. A Cayley graph for F 2 × F 2 which is not minimally almost convex. International Journal of Algebra and Computation, 2022, 32 (1), pp.115-126. ⟨10.1142/S0218196722500059⟩. ⟨hal-03886498⟩
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