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We consider the SOS (solid-on-solid) model, with spin values $0,1,2$, on theCayley tree of order two (binary tree). We treat both ferromagnetic andantiferromagnetic coupling, with interactions which are proportional to theabsolute value of the spin differences.  We present a classification of all translation-invariant phases (splittingGibbs measures) of the model: We show uniqueness in the case ofantiferromagnetic interactions, and existence of up to seven phases in the caseof ferromagnetic interactions, where the number of phases depends on theinteraction strength.  Next we investigate whether these states are extremal or non-extremal in theset of all Gibbs measures, when the coupling strength is varied, whenever theyexist.  We show that two states are always extremal, two states are alwaysnon-extremal, while three of the seven states make transitions betweenextremality and non-extremality. We provide explicit bounds on those transitionvalues, making use of algebraic properties of the models, and an adaptation ofthe method of Martinelli, Sinclair, Weitz.

Extremality of translation-invariant phases for a finite-state SOS-model on the binary tree