Existence of Invariant Planes in a Complex Projective 3-Space under Discrete Projective Transformation Groups

Abstract

Let $\Gamma$ be a finitely generated discrete subgroup of $\mathrm{PGL}(4,\mathbf{C})$ acting on $\mathbf{P}^3$. Suppose that $\Gamma$ leaves invariant a surface in $\mathbf{P}^3$. Then, except for a few cases, we can find a plane which is invariant by a finite index subgroup of $\Gamma$. The exceptional cases will be determined explicitly.