Residual properties of free products

TL;DR

This paper establishes conditions under which free products of residually C groups remain residually C, and explores residual properties of special HNN extensions and amalgamated free products, with implications for amenable and LEA groups.

Contribution

It provides new criteria for residual properties in free products and characterizes residual properties of certain HNN extensions and amalgamated free products.

Findings

Residually amenable groups are closed under free products.

LEA groups are closed under free products.

Characterization of residual properties in HNN extensions and amalgamated free products.

Abstract

Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for locally embeddable into C groups. As a corollary, we obtain that the class of residually amenable groups and the one of LEA groups (or initially subamenable groups in the terminology of Gromov) are closed under taking free products. Moreover, we consider the pro-C topology and we characterize special HNN extensions and amalgamated free products that are residually C, where C is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.