Rich groups, weak second order logic, and applications

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.

Regular bi-interpretability of Chevalley groups over local rings

“…A new round of development of this topic has appeared recently in the papers of A. Nies ([37], [36], etc), N. Avni, A. Lubotzky, C. Meiri ( [4], [5], also see [22]), A. Myasnikov, O. Kharlampovich and M. Sohrabi ( [25], [35], [34], etc. ), D. Segal and K. Tent ( [38]), B. Kunyavskii, E. Plotkin, N. Vavilov ( [28]) and others.…”

“…Khelif [26] (see also [25]) realised that one can use bi-interpretability of a finitely generated structure A with (Z, +×) as a general method to prove that A is QFA. Somewhat later, Scanlon independently used this method to show that each finitely generated field is QFA.…”