Residual properties of graph products of groups

TL;DR

This paper proves that certain residual properties of groups are preserved under graph products, especially focusing on residually amenable groups and local embeddability into amenable groups, expanding understanding of group constructions.

Contribution

It establishes that the class of residually C groups remains closed under graph products under specific conditions, including for residually amenable groups.

Findings

Residually C groups are closed under graph products if C is closed under subgroups, finite direct products, and free-by-C groups.

Graph products of residually amenable groups are residually amenable.

Local embeddability into amenable groups is preserved under graph products.

Abstract

We prove that the class of residually C groups is closed under taking graph products, provided that C is closed under taking subgroups, finite direct products and that free-by-C groups are residually C. As a consequence, we show that local embeddability into various classes of groups is stable under graph products. In particular, we prove that graph products of residually amenable groups are residually amenable, and that locally embeddable into amenable groups are closed under taking graph products.