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Abstract Using the theory of $${\mathbf {FS}} {^\mathrm {op}}$$ modules, we study the asymptotic behavior of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ , the Deligne–Mumford compactification of the moduli space of curves, for $$n\gg 0$$ . An $${\mathbf {FS}} {^\mathrm {op}}$$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of $${\overline {\mathcal {M}}_{g,n}}$$ the structure of an $${\mathbf {FS}} {^\mathrm {op}}$$ module and bound its degree of generation. As a consequence, we prove that the generating function $$\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$$ is rational, and its denominator has roots in the set $$\{1, 1/2, \ldots, 1/p(g,i)\},$$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ into irreducible $$\mathbf {S}_n$$ representations.
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SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES
Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics.
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- Award ID(s):
- 2003528
- PAR ID:
- 10221480
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 8
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2019.49
