Amenable coverings of complex manifolds and holomorphic probability measures

this paper we will demonstrate a direct connection between efficiencyof the push-forward and amenability of the covering . The latter is apurely combinatorial property of the pair of groups 1 (Y ) ae 1 (X); thecovering is amenable if there exists a 1 (X)-invariant finitely additive probabilitymeasure on the coset space 1 (X)= 1 (Y ). Amenability is discussedin more detail in x3.