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The Arithmetic is interpreted in all the groups of Richard Thompson andGraham Higman, as well as in other groups of piecewise affine permutations ofan interval which generalize the groups of Thompson and Higman. In particular,the elementary theories of all these groups are undecidable. Moreover,Thompson's group $F$ and some of its generalizations interpret the Arithmeticwithout parameters.

Interpretation of the Arithmetic in certain groups of piecewise affine permutations of an interval