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We show that the Yang-Baxter equations for two dimensional models admit as agroup of symmetry the infinite discrete group $A_2^{(1)}$. The existence ofthis symmetry explains the presence of a spectral parameter in the solutions ofthe equations. We show that similarly, for three-dimensional vertex models andthe associated tetrahedron equations, there also exists an infinite discretegroup of symmetry. Although generalizing naturally the previous one, it is amuch bigger hyperbolic Coxeter group. We indicate how this symmetry can help toresolve the Yang-Baxter equations and their higher-dimensional generalizationsand initiate the study of three-dimensional vertex models. These symmetries arenaturally represented as birational projective transformations. They maypreserve non trivial algebraic varieties, and lead to proper parametrizationsof the models, be they integrable or not. We mention the relation existingbetween spin models and the Bose-Messner algebras of algebraic combinatorics.Our results also yield the generalization of the condition $q^n=1$ so oftenmentioned in the theory of quantum groups, when no $q$ parameter is available.Comment: 23 page

Rational mappings, arborescent iterations, and the symmetries of integrability