# The structure of extended function groups

## TL;DR

This paper establishes a structural decomposition theorem for extended function groups, generalizing Maskit's decomposition for function groups to include orientation-reversing elements.

## Contribution

It provides the first formal statement and proof of a decomposition structure for extended function groups, extending classical results to a broader class.

## Key findings

- Decomposition theorem for extended function groups proved
- Extension of Klein-Maskit combination theorems to include orientation-reversing elements
- Structural understanding of extended Kleinian groups achieved

## Abstract

A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing elements and keep invariant a connected components of its region of discontinuity. An structural decomposition of function groups, in terms of the Klein-Maskit combination theorems, was provided by Maskit in the middle of the 70's. One should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. The aim of this paper is to state and procvide a proof of such a decomposition structural picture.