Exponential volumes of moduli spaces of hyperbolic surfaces

Mathematics > Algebraic Geometry [Submitted on 3 Nov 2024 (v1), last revised 18 Jun 2026 (this version, v2)] Title:Exponential volumes of moduli spaces of hyperbolic surfaces View PDF HTML (experimental)Abstract:A decorated surface S is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. This space carries a volume form. Let us fix the set K of distances between the horocycles at the adjacent cusps, and the set L of lengths of boundary circles without cusps. We get a subspace M(S; K,L) with the induced volume form Vol(K,L). However, if the cusps are present, the volume of the space M(S; K,L) is infinite. We introduce the exponential volume form exp(-W)Vol(K,L), where W is a positive function on the moduli space, given by the sum over cusps of the hyperbolic areas enclosed between the cusp and the horocycle at the cusp. We prove that the exponential volume, defined as the integral of the exponential volume form over the moduli space M(S; K,L), is always finite. We suggest that the moduli spaces M(S; K,L) with the exponential volume forms are the true analogs of the classical moduli spaces of Riemann surfaces, with the Weil-Petersson volume forms. In particular, they should be relevant to the open string theory. We support this by proving an unfolding formula for the integrals of measurable functions multiplied by the exponential volume form. It expresses them as finite sums of similar integrals over moduli spaces for simpler surfaces. They generalise Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We show that exponential volumes for elementary decorated surfaces give rise to a commutative algebra E, which we call the positive Hecke-Whittaker algebra for PGL(2,R). Exponential volumes for all decorated surfaces and unfolding formulas extend the algebra E to all decorated surfaces. Submission history From: Alexander Goncharov [view email][v1] Sun, 3 Nov 2024 16:03:36 UTC (348 KB) [v2] Thu, 18 Jun 2026 08:37:20 UTC (343 KB) Current browse context: math.AG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.