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Invariant and stationary measures for the action on Moduli space

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Abstract

We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold.

The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Alex Eskin

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Maryam Mirzakhani

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  1. Alex Eskin
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  2. Maryam Mirzakhani
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Correspondence to Alex Eskin.

Additional information

Research of the first author is partially supported by NSF grants DMS 0604251, DMS 0905912 and DMS 1201422.

Research of the second author is partially supported by the Clay foundation and by NSF grant DMS 0804136.

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Eskin, A., Mirzakhani, M. Invariant and stationary measures for the action on Moduli space. Publ.math.IHES 127, 95–324 (2018). https://doi.org/10.1007/s10240-018-0099-2

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