Fixed-point free actions on compact surfaces (Q1365451)

Let \(G_R\) (\(G_C\)) be the group of affine transformations of the real (complex) line; and \(G_\alpha\), \(\alpha= a+ ib \in C\), be the three-dimensional Lie group with trivial center whose Lie algebra is given by the structure equations: \([X_1, Y]= aX_1+ bX_2\), \([X_2, Y]= -bX_1+ aX_2\), \([X_1, X_2]= 0\). The main result of the paper under review is the following theorem: a Lie group \(G\) admits a continuous action without fixed points on a compact surface \(M\) with Poincaré-Euler characteristic \(\chi\) if and only if \(G\) has the following quotient groups: (1) for \(\chi < 0\): \(G_R\); (2) for \(\chi= 0\): the circle \(S^1\) or \(PSL(2, R)\); (3) for \(\chi > 0\): \(G_R\), \(G_C\), \(G_\alpha\), \(PSL(2, R)\); and \(PSO(3)\), \(PSL(3,R)\), \(PSL(2,C)\) in case \(M\) is without boundary.