  It is shown that timelike surfaces of constant mean curvature 1 in anti-de
Sitter 3-space can be constructed from a pair of Lorentz holomorphic and
Lorentz antiholomorphic null curves in PSL(2,R) via Bryant type representation
formulae. These formulae are used to investigate an explicit one-to-one
correspondence, the so-called Lawson correspondence, between timelike surfaces
of constant mean curvature 1 in anti-de Sitter 3-space and timelike minimal
surfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces in
anti-de Sitter 3-space, which is a close analogue of the classical Gauss map is
considered. It is discussed that the hyperbolic Gauss map plays an important
role in the study of timelike surfaces of constant mean curvature 1 in anti-de
Sitter 3-space. In particular, the relationship between the Lorentz
holomorphicity of the hyperbolic Gauss map and timelike surfaces of constant
mean curvature 1 in anti-de Sitter 3-space is studied.
