Rich groups, weak second order logic, and applications

TL;DR

This paper explores first-order rich groups, which have the same logical power as weak second order logic, revealing many finitely generated examples and their properties, with implications for Malcev's problems.

Contribution

It introduces the concept of rich groups, provides methods to identify them, and analyzes their properties and applications in group theory.

Findings

Many finitely generated rich groups exist

Rich groups are between hyperbolic and nilpotent groups

Methods to prove groups are rich are developed

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.