Chern--Simons Gravity from 3+1 Dimensional Gravity

In the context of a Poincar'e gauge theoretical formulation, pure gravity in 3+1-dimensions is dimensionally reduced to gravity in 2+1-dimensions with or without cosmological constant $\mathrm{\Lambda }$. The dimensional reductions are consistent with the gauge symmetries, mapping ISO(3, 1) gauge transformations into ISO(2,1) ones. One of the reductions leads to Chern-Simons-Witten gravity. The solutions of 2+1-gravity with $\mathrm{\Lambda }\le 0$ (in particular the black-hole solution recently found by Banados, Teitelboim and Zanelli) and those of 1+1-dimensional Liouville gravity, are thus mapped into 3+1-dimensional vacuum solutions.