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Virtual homological spectral radii for automorphisms of surfaces

ABSTRACT: In this talk, I will report some recent progress on virtual properties of surface automorphisms. Here a surface automorphism refers to any orientation-preserving self-homeomorphism of a compact orientable surface, and a virtual property refers to a property that holds up to lifting to some finite covering space. It has been conjectured by C. T. McMullen that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane. I will give a general introduction to the background on this topic, and discuss some ideas to prove it.