Fixed points of analytic actions of supersoluble Lie groups on compact surfaces (Q2773065)

The problem of finding conditions on solvable group actions that guarantee existence of a fixed point is studied. It is well known that every flow (i.e. an action of \(\mathbb R\)) on a compact connected surface has a fixed point, provided the Euler characteristic does not vanish. Lima extended this result to any abelian group, and Plante to any nilpotent group. On the other hand there exist fixed-point-free actions on compact surfaces of the solvable group of homeomorphisms of \(\mathbb R\) of the form \(x\mapsto ax+b\), \(a>0\), \(b\in\mathbb R\). The main theorem of the paper under review states that every analytic action of a connected supersoluble Lie group on a compact surface with non-zero Euler characteristic possesses a fixed point. The authors present also an example showing that supersolubility is indispensable even in the real-analytic category.