# Two-ended quasi-transitive graphs

## TL;DR

This paper explores the structure of two-ended quasi-transitive graphs, providing a new way to decompose them over finite subgraphs and characterizing groups acting on these graphs with finitely many orbits.

## Contribution

It introduces a novel splitting method for two-ended quasi-transitive graphs that do not have dominated ends, extending the classical group-theoretic characterizations.

## Key findings

- New splitting method for two-ended quasi-transitive graphs
- Characterization of groups acting with finitely many orbits on these graphs
- Extension of classical group decomposition results to graph structures

## Abstract

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation $A\ast_C B$ or an HNN-extension $\ast_{\phi} C$, where $C$ is a finite group and $[A:C]=[B:C]=2$ and $\phi\in Aut(C)$. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.