Lineal gravity from planar gravity.

We show how to obtain the two dimensional black hole action by a dimensional reduction of the three dimensional Einstein action with a non-zero cosmological constant. Starting from the Chern-Simons formulation of 2+1 gravity, we obtain the 1+1-dimensional gauge formulation given by Verlinde. Remarkably, the proposed reduction shares the relevant features of the formulation of Cangemi and Jackiw, without the need for a central charge in the algebra. We show how the Lagrange multipliers in these formulations appear naturally as the remnants of the three-dimensional connection associated to symmetries that have been lost in the dimensional reduction. The proposed dimensional reduction involves a shift in the three dimensional connection whose effect is to make the length of the extra dimension infinite.