A representation of range decreasing group homomorphisms
Ning Zhang, Lifan Liu
TL;DR
This work characterizes range decreasing group homomorphisms on full mapping spaces $\mathcal{F}(V,G)$ (and $\mathcal{F}_c(V,G)$) by showing that any such homomorphism factors as an evaluation at a point $E_{\bar{v}}$, possibly composed with a central homomorphism $\psi$ on the quotient by the compactly supported subgroup $\mathcal{F}^0_c(V,G)$. The central result, Theorem general, yields $f(x)=\psi([x])\,x(\bar{v})$, and reduces to $f=E_{\bar{v}}$ when $Z(G)$ is trivial. The paper also provides equivalence conditions, analyzes cases for tori and $S^1$, and extends to mappings between mapping spaces, with implications for weighted composition operators and automatic continuity in various topological settings. These contributions unify and extend prior results on range decreasing homomorphisms, enabling explicit computation of a broad class of homomorphisms and clarifying their structure across different target groups and domains.
Abstract
The method of range decreasing group homomorphisms can be applied to study various maps between mapping spaces, includin holomorphic maps, group homomorphisms, linear maps, semigroup homomorphisms, Lie algebra homomorphisms and algebra homomorphisms [Z1, Z2]. Previous studies on range decreasing group homomorphisms have primarily focused on specific subsets of mapping groups. In this paper, we provide a characterization of a general range decreasing group homomorphism applicable to the entire mapping group. As applications, we compute a particular class of homomorphisms between mapping groups and identify all range decreasing group homomorphisms defined on specific mapping groups.