Affine laminations and their stretch factors

A lamination L embedded in a manifold M is an affine lamination if its lift L to the universal cover M of M is a measured lamination and each covering translation multiplies the measure by a factor given by a homomorphism, called the stretch homomorphism, from π₁(M) to the positive real numbers. There is a method for analyzing precisely the set of affine laminations carried by a given branched manifold B embedded in M. The notion of the "stretch factor" of an affine lamination is a generalization of the notion of the stretch factor of a pseudo-Anosov map. The same method that serves to analyze the affine laminations carried by B also allows calculation of stretch factors. Affine laminations occur commonly as essential 2-dimensional laminations in 3-manifolds. We shall describe some examples. In particular, we describe affine essential laminations which represent classes in real 2-dimensional homology with twisted coefficients.