# A Limit Set Intersection Theorem for Graph of Relatively Hyperbolic Groups

## TL;DR

This paper proves a limit set intersection theorem for relatively hyperbolic groups structured as a finite graph, extending concepts from hyperbolic groups to a broader class with quasi-isometric embeddings.

## Contribution

It establishes the limit set intersection property for conjugates of vertex and edge groups in relatively hyperbolic groups with a graph decomposition.

## Key findings

- Proves limit set intersection property for relatively hyperbolic groups
- Extends Sardar's work from hyperbolic to relatively hyperbolic groups
- Provides a new understanding of the geometric structure of these groups

## Abstract

Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and edge groups satisfy the limit set intersection property for conical limit points. This result is motivated by the work of Sardar for graph of hyperbolic groups.