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

TL;DR

This paper explores the interpretation of arithmetic within various groups of piecewise affine permutations, including Thompson and Higman groups, revealing their elementary theories are undecidable and showing some groups interpret arithmetic without parameters.

Contribution

It extends the understanding of arithmetic interpretation to a broader class of groups, including generalizations of Thompson and Higman groups, and establishes undecidability results.

Findings

Elementary theories of these groups are undecidable.

Thompson's group F and some generalizations interpret arithmetic without parameters.

Arithmetic interpretation applies to a wide class of piecewise affine permutation groups.

Abstract

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.