Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

2025/07/14 by Paul Apisa, Apisa, Paul, Nick Salter +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · pdf · doi:10.48550/arxiv.2507.10685

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Connected components of strata of residueless meromorphic differentials
• Teichmüller curves in hyperelliptic components of meromorphic strata
• Period realization of meromorphic differentials with prescribed invariants
• Moduli spaces of complex affine and dilation surfaces
• Closed geodesics in dilation surfaces
• Horizon saddle connections and Morse-Smale dynamics of dilation surfaces
• On limit sets for geodesics of meromorphic connections
• Framed mapping class groups and the monodromy of strata of Abelian differentials
• SL₂(ℝ)-dynamics on the moduli space of one-holed tori
• Borel's stable range for the cohomology of arithmetic groups
• Teichm "uller dynamics, dilation tori and piecewise affine circle\n homeomorphisms
• Mapping class group dynamics and the holonomy of branched affine\n structures
• Mapping class group dynamics on Aff(C)-characters
• Connected components of the strata of the moduli space of meromorphic differentials

Related

• A survey of conformally invariant measures on H^m(δ)
• A Superficial Working Guide to Deformations and Moduli
• K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra
• Horocycle dynamics in rank one invariant subvarieties I: weak measure classification and equidistribution
• A twisted invariant of a compact Riemann surface