For a group with one end does the property of connected spheres follow? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:02:06Z http://mathoverflow.net/feeds/question/101284 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101284/for-a-group-with-one-end-does-the-property-of-connected-spheres-follow For a group with one end does the property of connected spheres follow? nikutaibi 2012-07-04T05:23:26Z 2012-07-04T07:50:09Z <p>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.</p> <p>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 $G$ <em>has one end</em> 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 $y$.</p> <p>For a group with one end let us define a property, that rises from the studies of group actions on the circle, namely <em>a property of connected spheres</em>: 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.</p> <p>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.</p> <p>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?</p>