Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911055527018496 |
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| author | Apisa, Paul Salter, Nick |
| author_facet | Apisa, Paul Salter, Nick |
| contents | The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of Masur-Veech measure, i.e. an SL(2,R)-invariant Lebesgue class measure on strata or explicit covers thereof. This relies on a novel computation of cohomology with coefficients for the mapping class group. The computation produces a framed mapping class group invariant measure on representation varieties that naturally appear as the codomains of the periods maps that coordinatize strata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10685 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces Apisa, Paul Salter, Nick Geometric Topology Dynamical Systems The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of Masur-Veech measure, i.e. an SL(2,R)-invariant Lebesgue class measure on strata or explicit covers thereof. This relies on a novel computation of cohomology with coefficients for the mapping class group. The computation produces a framed mapping class group invariant measure on representation varieties that naturally appear as the codomains of the periods maps that coordinatize strata. |
| title | Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces |
| topic | Geometric Topology Dynamical Systems |
| url | https://arxiv.org/abs/2507.10685 |