Isometry groups of spaces with lower ricci curvature bounds

Isometric actions on Riemannian manifolds have been a usefu tool to investigate the interaction between the topology and the Riemannian metric a manifold might admit. A major result in the area is the theorem of Myers-Steenrod [4] stating that the isometry group of a Riemannian manifold is a Lie group. Later this result has been extended to a wider class of spaces with certain synthetic notions of curvature bounds such as Alexandrov spaces by Fukaya-Yamaguchi [2] and limits of manifolds with lower Ricci curvature bounds by Cheeger-Colding [1]. The aim of this talk is to study the isometry group of a RCD(K;N)-space, that is a

metric measure space with a notion of Ricci curvature in terms of the optimal transport of probability measures and the convexity of an entropy functional. Additionally we obtain bounds on the dimension of the group and a rigidity result when the dimension is maximal. This is joint work with Dr. Luis Guijarro [3].

References

[1] Cheeger, J.; Colding, T. H.,

On the structure of spaces with Ricci curvature bounded below. II.

J. Di erential Geom. 54 (2000), no. 1, 1335.

[2] Fukaya, K.; Yamaguchi, T.,

Isometry groups of singular spaces.

Math. Z. 216 (1994), no. 1, 31-44.

[3] Guijarro, L.; Santos-Rodrguez, J.,

On the isometry group of RCD (K;N)-spaces arXiv:1608.06467 [math.DG]

[4] Myers, S. B.; Steenrod, N. E.

The group of isometries of a Riemannian manifold.

Ann. of Math. (2) 40 (1939), no. 2, 400-416.