On groups with weak Sierpiński subsets

This is joint work with Y. Cornulier and P. Słanina.
Let G be an infinite group. A subset E of  G is called a weak Sierpiński subset if there exist two elements g, h in G and two distinct points a and b in E  such that  gE=E \{a} and hE=E\{b}. In the talk we discuss the subgroup H=<g,h> and provide its presentation if it is not free over (g,h). We also characterize all weak Sierpiński subsets in groups with such a presentation.