The profinite completion of accessible groups

TL;DR

This paper studies the profinite completion of a specific class of accessible, residually finite groups, showing it nearly captures their JSJ-decomposition and calculating the genus for free products within this class.

Contribution

It introduces a new class of groups with restrictions on their JSJ-decompositions and analyzes how their profinite completions reflect their structure.

Findings

Profinite completion nearly detects JSJ-decomposition.

Computed the genus of free products of groups in the class.

Established properties of residually finite accessible groups.

Abstract

We introduce a class $\A$ of finitely generated residually finite accessible groups with some natural restriction on one-ended vertex groups in their JSJ-decompositions. We prove that the profinite completion of groups in $\A$ almost detects its JSJ-decomposition and compute the genus of free products of groups in $\A$.