Separating cyclic subgroups in graph products of groups

TL;DR

This paper investigates the preservation of cyclic subgroup separability in graph products of groups, extending the analysis to pro-p topologies and characterizing p-isolated cyclic subgroups in right-angled Artin groups.

Contribution

It proves that cyclic subgroup separability is preserved under graph products and develops tools for analyzing p-isolated cyclic subgroups in pro-p topologies.

Findings

Cyclic subgroup separability is preserved in graph products.

p-isolated cyclic subgroups are closed in pro-p topology for many groups.

Characterization of p-isolated cyclic subgroups in right-angled Artin groups.

Abstract

We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-$p$ case. For a wide class of groups we show that the relevant cyclic subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology of the graph product. In particular, we show that every $p$-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and we fully characterise such subgroups.