[Paper Review] The isometry group of an $\mathsf{RCD}^*$-space is Lie

Gerardo Sosa|arXiv (Cornell University)|Sep 7, 2016

Geometric Analysis and Curvature Flows23 references3 citations

TL;DR

This paper establishes that the isometry group and measure-preserving isometry group of an RCD$^*$-space are Lie groups under mild geometric and measure-theoretic conditions. By analyzing tangent cones and optimal transport properties, the author proves that RCD$^*$-spaces—generalizing Ricci limit and weighted Riemannian spaces—satisfy the necessary and sufficient conditions for smooth automorphism groups, resolving a key question in metric measure geometry.

ABSTRACT

We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD*-spaces and, under extra assumptions, by CD-spaces, CD*-spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups. More generally we show that spaces with good optimal transport properties meet as well the hypotheses.

Motivation & Objective

• To determine necessary and sufficient conditions under which the isometry group and measure-preserving isometry group of a metric measure space are Lie groups.
• To investigate whether curvature-dimension conditions such as RCD$^*$, CD, CD$^*$, and MCP imply smoothness of isometry groups.
• To clarify the role of optimal transport and tangent cone structure in ensuring Lie group structure for automorphism groups.
• To show that the MCP condition alone is insufficient to guarantee Lie group structure, even when other curvature bounds hold.
• To extend known results from Riemannian and Alexandrov spaces to broader classes of metric measure spaces with generalized Ricci curvature bounds.

Proposed method

• Introduces a condition (a) involving uniform bounds on fixed point sets of nontrivial isometries within a small ball, ensuring effective separation of group elements.
• Applies the theory of tangent cones, assuming m-a.e. Euclidean or well-behaved (e.g., Carnot group) tangent cones to control local group behavior.
• Uses optimal transport techniques to analyze measure-preserving isometries, particularly in spaces with curvature-dimension bounds.
• Applies results from the literature on doubling measures and unique tangents (e.g., Le Donne’s theorem) to extend the framework beyond Riemannian geometry.
• Reduces the analysis of higher-dimensional necklaces to lower-dimensional cases via projection and recursive transport arguments.
• Employs a recursive geodesic transport construction across multiple 'diamonds' in a necklace structure to verify the MCP(2,3) condition in 2-necklaces.

Experimental results

Research questions

• RQ1 Under what conditions is the isometry group of a metric measure space a Lie group?
• RQ2 Do RCD$^*$-spaces, which generalize Ricci limit and weighted Riemannian spaces, have Lie isometry groups?
• RQ3 Is the measure contraction property (MCP) sufficient to ensure that the isometry group is a Lie group?
• RQ4 Can the smoothness of automorphism groups be guaranteed by optimal transport properties and tangent cone regularity?
• RQ5 What is the role of the fixed point set condition (a) in ensuring Lie group structure for isometry groups?

Key findings

• The isometry group and measure-preserving isometry group of an RCD$^*$-space are Lie groups, as proven in Corollary 1.2.
• The condition (a) in Theorem 1.1—bounding fixed point sets uniformly in a ball—is both necessary and sufficient for the isometry group to be a Lie group.
• The MCP condition alone is insufficient to guarantee a Lie group structure, as shown by the counterexample of the infinite necklace space with $ ext{ISO}( ext{FN}) = ext{ISO}_{rak{m}}( ext{FN}) = igoplus_{ u=1}^ u igackslash \{ \pm 1 \}$.
• The proof of the MCP(2,3) condition for 2-necklaces relies on a recursive geodesic transport construction across three regions: from $ ilde{x}$ to $ ilde{x}$, through $M^1$, and to $A_{x'}$, preserving relative height ratios.
• The density estimate $rac{drak{n}_t}{drak{m}}( u_t) \leq \frac{1}{t^2} \frac{h(x_t,D_1)}{h(x,D_1)} \frac{drak{n}_1}{drak{m}}(\nu_1)$ is used to verify the MCP inequality in the first segment.
• For $t \in [\hat{t},1]$, the relative density is constant in $y$-coordinate, reducing the problem to the 0-necklace case, which is already known to satisfy MCP(2,3).