  We study the Teichm\"uller metric on the Teichm\"uller space of a surface of
finite type, in regions where the injectivity radius of the surface is small.
The main result is that in such regions the Teichm\"uller metric is
approximated up to bounded additive distortion by the sup metric on a product
of lower dimensional spaces. The main technical tool in the proof is the use of
estimates of extremal lengths of curves in a surface based on the geometry of
their hyperbolic geodesic representatives.
