Length functions exponentially distorted on subgroups of complex Lie groups

We introduce a notion of a length function exponentially distorted on a (compactly generated) subgroup of a locally compact group. We prove that for a connected linear complex Lie group there is a maximum equivalence class of length functions exponentially distorted on a normal integral subgroup lying between the exponential and nilpotent radicals. Moreover, a function in this class admits an asymptotic decomposition similar to that previously found by the author for word length functions, i.e., in the case of exponential radical (J Lie Theory 29:4, 1045–1070, 2019). In the general case we use auxiliary length functions constructed via holomorphic homomorphisms to Banach PI-algebras.