Farrell-Jones via Dehn fillings

TL;DR

This paper extends the Dehn filling theorem to analyze the structure of pre-images of infinite order elements and applies it to prove the Farrell-Jones conjecture for certain relatively hyperbolic groups.

Contribution

It generalizes the Dehn filling theorem and demonstrates the Farrell-Jones conjecture for groups hyperbolic relative to residually finite groups.

Findings

Pre-images of infinite order elements have a free product structure.

Groups hyperbolic relative to residually finite groups satisfy the Farrell-Jones conjecture.

Extension of Dehn filling techniques to broader classes of groups.

Abstract

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farrell-Jones conjecture, they satisfy the Farrell-Jones conjecture.