Arithmetic in groups of piece-wise affine permutations of an interval

Bardakov and Tolstykh have recently shown that Richard Thompson's group

$F$ interprets the Arithmetic $(\mathbb Z,+,\times)$ with parameters. We

consider a class of infinite groups of piecewise affine permutations of

an interval which contains all the three groups of Thompson and some

classical families of finitely presented infinite simple groups. We have

interpreted the Arithmetic in all the groups of this class. In particular

we have obtained that the elementary theories of all these groups are

undecidable. Additionally, we have interpreted the Arithmetic in $F$ and

some of its generalizations without parameters.

This is a joint work with Tuna Altınel.