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5 corrected the definition of a property of connected spheres after the remark of M. Sapir; added 17 characters in body

The

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$. 4 added 61 characters in body One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group$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$in terms of generators. The definition that has risen from the studies of group actions on the circle is a specific group property, namely a property of connected spheres: a group has a property of connected spheres if there exists$C>0$such that for any sphere$S_R$of radius$R$the points in the fiber between$S_R$and$S_{R+C}$could be connected by the path in the group, i.e. for any$x,y$such that$R<|x|< R+C$and$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$and all the steps still lie in the fiber between the spheres. Let me recall a classical definition of a group with one end. If$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$such that for any$x,y \in G \setminus L$there is a path in$G \setminus K$joining$x$and$y$. The question is if for a group with one end the property of connected spheres holds automatically or not, and what are the examples in the case? 3 given a definition of a group with one end; added 1 characters in body One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group$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$in terms of generators. The definition that has risen from the studies of group actions on the circle is a specific group property, namely a property of connected spheres: a group has a property of connected spheres if there exists$C>0$such that for any sphere$S_R$of radius$R$the points in the fiber between$S_R$and$S_{R+C}$could be connected by the path in the group, i.e. for any$x,y$such that$R<|x|< R+C$and$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$/. Let me recall a classical definition of a group with one end. If$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\$.

The question is if for a group with one end this the property of connected spheres holds automatically or not, and what are the examples in the case?

2 corrected spelling; added 2 characters in body
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