In this talk, we introduce the ends of graphs, and then the ends of groups, which are the ends of Cayley graphs of groups. It is proven that the number of ends of group G, e(G) does not depend on the Cayley graphs, and hence, it can be studied independently of generating set. We investigate free ends of graphs and group. We prove that a group G has free end if and only if it does not have more than 2 ends. As a consequence, we show that a free group has free end if and only if G∼= Z.

Keywords