Rich groups, weak second order logic, and applications
Logic
2022-10-18 v2 Group Theory
Abstract
In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.
Cite
@article{arxiv.2109.13133,
title = {Rich groups, weak second order logic, and applications},
author = {Olga Kharlampovich and Alexei Myasnikov and Mahmood Sohrabi},
journal= {arXiv preprint arXiv:2109.13133},
year = {2022}
}
Comments
This a mainly expository paper, the final version of which appeared as a contribution to the book "Groups and Model Theory, GAGTA Book 2", edited by Kharlampovich and Sklinos, published in 2021 by de Gruyter (Misprints are corrected in this version)