A geometric classification of Lie groups (Q1573714)

This paper is part of a program aimed at the study of Lie groups by way of heat diffusion on the group. To describe the flavor of the results, let \(G\) be a connected Lie group with Lie algebra \({\mathbf g}\), and consider the left Haar measure, denoted \(dx\), on \(G\). Let \(\Omega\) be a neighborhood of the identity \(e\) such that \(\Omega= \Omega^{-1}\), and define \(\gamma (n)=\) Haar measure \((\Omega^n)\) for \(n\geq 1\). Let \(\varphi\in C_0(G)\), and consider the measure \(d\mu(x)= \varphi(x)dx\), where \(\varphi\) is chosen such that \(\mu\) is a probability measure and the inverse map preserves \(\mu\). Let \(\varphi_n\) be the function such that \(d\mu^{*n}(x)= \varphi_n(x)dx\), where \(\mu^{*n}\) denotes the convolution of \(\mu\) with itself \(n\) times. Fix \(g\in G\), and define \(\varphi(n)=\varphi_n(g)\). Then the unimodular Lie groups can be separated into two disjoint classes. This classification can be achieved (i) geometrically, essentially by the condition that \(\gamma(n)\) grows exponentially or not, (ii) algebraically, by a condition on the Lie algebra known as the \(R\)-condition, and (iii) analytically, essentially by the condition that \(\varphi(n) =O(e^{-cn^{1 \over 3}})\) for some \(c>0\) or not. However, these conditions are insufficient to provide a dichotomy for general connected (non-unimodular) Lie groups, for which the situation is more subtle. Previous work of the author and collaborators has identified the appropriate algebraic and analytical properties to extend this classification to arbitrary connected Lie groups. The present paper completes this classification scheme by identifying the geometric characterization of the dichotomy. The author begins the paper with a very well written overview of the field and a guide to the reader, both of which make this paper much easier to digest and appreciate.