The non-unimodular thick/thin dichotomy for connected Lie groups

Let $G$ be a connected Lie group. The notions of algebraic and analytic thickness and thinness are those used in the paper, and Theorem is the thick/thin dichotomy for unimodular connected Lie groups. Non-unimodular thick/thin dichotomy. Theorem holds in full generality, without assuming that $G$ is unimodular. This extends the paper's dichotomy from unimodular to arbitrary connected Lie groups. The authors point to non-unimodular semidirect products as test cases, but no resolution is given.