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Let me recall a classical definition that of a group with one end. If $G$ is a connected, locally path connected, locally compact topological space. Then $G$ has risen one end if given a compact subset $K \subset G$, there is a compact set $L$: $K \subset L$ suchthat for any $x,y \in G \setminus L$ there is a path in $G \setminus K$ joining $x$ and $y$.
For a group with one end let us define a property, that rises from the studies of group actions on the circleis a specific group property, namely a property of connected spheres: For any ball $B_R$ let us take a nonbounded component of its compliment $(B_R)^c_{\infty}$: this component is unique since our group has one end.
Then we say, that a group has a property of connected spheres if there exists $C>0$ such that for any sphere ball $S_R$ B_R$ of radius $R$ the points in the fiber between $S_R$ and (B_R)^c_{\infty} \cap B_{R+C} $ S_{R+C}$ could be connected by the path in the group, i.e. for any $x,y$ such that in the fiber $R<|x|< R+C$ and (B_R)^c_{\infty} \cap B_{R+C} $ R<|y|< R+C$ there exists a finite number of group elements, such that $x=gy$ and $g$ is a word in the alphabet $g_1, \ldots g_n$ and all the steps still lie in the fiber considered between the spheres.
Let me recall a classical definition of a group with one end.If $G$ is a connected, locally path connected, locally compact topological space. Then $G$ has one end if given a compact subset $K \subset G$, there is a compact set $L$: $K \subset L$ suchthat for any $x,y \in G \setminus L$ there is a path in $G \setminus K$ joining $x$ and $y$.
