A representation of range decreasing group homomorphisms

Abstract

The method of range decreasing group homomorphisms was first introduced in \cite{z23} and later generalized in \cite{z25}. This approach has proved effective for analyzing a broad class of weighted composition operators between mapping spaces, including holomorphic maps, group homomorphisms, linear maps, semigroup homomorphisms, Lie algebra homomorphisms and algebra homomorphisms. Previous work on range decreasing group homomorphisms has focused mainly on specific subsets of mapping groups. In this paper, we characterize range decreasing group homomorphisms on the entire mapping group. As applications, we identify all such group homomorphisms on certain mapping groups and classify a particular type of homomorphisms between mapping groups.