On trivial ends of Cayley graph of groups

‎In this paper, first we introduce the end of locally finite graphs as an equivalence class of infinite paths in the graph. Then we mention the ends of finitely generated groups using the Cayley graph. It was proved that the number of ends of groups are not depended on the Cayley graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For each of these numbers, some results have been obtained in the structure of groups, the most well-known of which is Stallings theorem providing the structure of groups with more one ends as the amalgamated free product or HNN extension. Specifically, it was proved that group with exactly two ends is a virtually Z group. After that, we introduce the trivial end of the graphs and show that the trivial end is exactly the same as the special type of infinite path. Finally, we prove that the existence of trivial end for Cayley graph of a group is equivalent to being a free group, and this implies that the Cayley graph of a group has a trivial end if and only if all of its ends are trivial.