Interpretation of the Arithmetic in certain groups of piecewise affine   permutations of an interval
  The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an 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 Arithmetic without parameters. 
Tuna Alt{\i}nel, Alexey Muranov