Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups

TL;DR

This paper establishes that for graph products of finite abelian groups, elementary equivalence coincides with isomorphism, and specifically for right-angled Coxeter groups, elementary equivalence implies they are identical structures.

Contribution

It proves that elementary equivalence and isomorphism are equivalent for graph products of finite abelian groups, including right-angled Coxeter groups.

Findings

Elementary equivalence coincides with isomorphism for these groups.

Two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.

The result characterizes the logical and structural similarity of these groups.

Abstract

We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\exists\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.