On groups elementarily equivalent to a group of triangular matrices $T_n(R)$

TL;DR

This paper characterizes groups that are elementarily equivalent to the group of invertible upper triangular matrices over various types of rings, providing conditions based on the ring's algebraic properties.

Contribution

It establishes necessary and sufficient conditions for a group to be elementarily equivalent to $T_n(R)$ over different algebraic structures.

Findings

Characterization over algebraically closed fields

Characterization over real closed fields

Characterization over rings of integers of number fields

Abstract

In this paper we investigate the structure of groups elementarily equivalent to the group $T_n(R)$ of all invertible upper triangular $n\times n$ matrices, where $n\geq 3$ and $R$ is a characteristic zero integral domain. In particular we give both necessary and sufficient conditions for a group being elementarily equivalent to $T_n(R)$ where $R$ is a characteristic zero algebraically closed field, a real closed field, a number field, or the ring of integers of a number field.