Interpr\'etation de l'Arithm\'etique dans certains groupes de permutations affines par morceaux d'un intervalle

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 some interpret arithmetic without parameters.

Contribution

It extends the interpretation of arithmetic to a broader class of groups of piecewise affine permutations, generalizing previous results on Thompson and Higman groups.

Findings

Elementary theories of these groups are undecidable

Thompson's group F interprets arithmetic without parameters

Generalizations of Thompson and Higman groups also interpret arithmetic

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.