We obtain a topological and weakly equivariant classification of closed threedimensional Alexandrov spaces with an effective, isometric circle action. This generalizes the topological and equivariant classifications of Raymond [26] and Orlik and Raymond [23] of closed three-dimensional manifolds admitting an effective circle action. As an application, we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.2010 Mathematics Subject Classification. Primary 53C23; Secondary 57M60, 57S25.

For n-dimensional Riemannian manifolds M with Ricci curvature bounded below by −(n − 1), the volume entropy is bounded above by n − 1. If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD * (−(N − 1), N) spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in RCD * spaces. As an application, we obtain an almost rigidity result which partially recovers a result by Chen-Rong-Xu for Riemannian manifolds.

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We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed 3-manifold with a local circle action and finitely many copies of the suspension of the real projective plane.

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