SPLITTINGS AND THE ASYMPTOTIC TOPOLOGY OF THE LAMPLIGHTER GROUP

It is known that splittings of one-ended finitely presented groups over 2-ended groups can be characterized geometrically. Here we show that this characterization does not extend to all finitely generated groups, by showing that the lamplighter group is coarsely separated by quasi-lines. It is also known that virtual surface groups are characterized in the class of one-ended finitely presented groups by the property that their Cayley graphs are coarsely separated by quasi-circles. Answering a question of Kleiner we show that the Cayley graph of the lamplighter group is coarsely separated by quasi-circles. It follows that the quasi-circle characterization of virtual surface groups does not extend to the finitely generated case.