Cyclic splittings of pro-$\mathcal{C}$ groups

TL;DR

This paper extends the Rips-Sela Theorems to pro- groups, demonstrating how such groups can be split over cyclic subgroups, with  being a class of finite groups with specific closure properties.

Contribution

It introduces a pro- version of the Rips-Sela Theorems, broadening the understanding of group splittings in the pro- setting.

Findings

Pro- groups can be split over cyclic subgroups under certain conditions.

The paper generalizes classical theorems to a broader class of groups.

Provides new tools for analyzing group actions in the pro- context.

Abstract

In this paper we proved a pro-$\mathcal{C}$ version of the Rips-Sela Theorems on splittings as an amalgamated free product or HNN-extension over a cyclic subgroup, where $\mathcal{C}$ is a class of finite groups closed for subgroups, quotients, finite direct products and extensions with abelian kernel.