In this book strong cosmic censorship and especially the following question is discussed: Let \(\Sigma\) be a compact (partial) Cauchy surface and \(G\) be a Lie group. Does the set of those vacuum Cauchy data which are invariant under the action of \(G\) contain a dense subset such that for each element of this subset the maximal extension of its Cauchy development is globally hyperbolic and satisfies Einstein’s equation for vacuum?
(a) If the Cauchy data are invariant under the action of a group of dimension \(\geq 3\), then the question can be answered, unless the spacetime in question is a Bianchi IX model. The answer depends on the group. (b) There are some partial results if the symmetry group is 2- dimensional \((U(1)\times U(1))\). The question can be affirmatively answered for polarized Gowdy spacetimes and for a certain six parameter family of non-polarized \(U(1)\times U(1)\) symmetric spacetimes. The main result of the book is the \(U(1)\times U(1)\) stability of the singularity of the \((2/3,2/3,-1/3)\)- Kasner metric. This is an affirmative answer to the question for one time direction in some neighbourhood of the Kasner solution. (c) The author also gives a review of related results (without proofs). (d) The book contains a discussion of maximal development criteria and a detailed proof that spacetime inherits the symmetries exhibited by the Cauchy data.