A rigidity theorem for group extensions

We consider the class of discrete groups which arise as fundamental groups of iterated surface fibrations; that is, of complexes obtained from a sequence of fibrations in which all bases and the initial fibre are hyperbolic surfaces. Group theoretically, this corresponds to studying the class of iterated extensions of hyperbolic surface groups. In 4], for the case of a single extension we conjectured and partially established that no group can arise from more than a finite number of such extensions. Here we show that the result holds in complete generality. As remarked in 4], the result has a strong affinity with the rigidity theorems of Parshin 7] and Arakelov 1] for fibred (complex) algebraic surfaces.