One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group $G$ with generators $g_1, \ldots g_n$. The notion of a length of an element $g$ can be given as a length of a minimal representation of $g$ in terms of generators.
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 $L$: $K \subset L$ such that 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 circle, 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 ball $B_R$ of radius $R$ the points in the fiber $(B_R)^c_{\infty} \cap B_{R+C} $ could be connected by the path in the group, i.e. for any $x,y$ in the fiber $(B_R)^c_{\infty} \cap B_{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.
The question is if for a group with one end the property of connected spheres holds automatically or not, and what are the examples in the case?
