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Équations différentielles motiviques et au–delà

Exponential volumes of moduli spaces of hyperbolic surfaces and recursions

IHP, Paris
salle 314

This is a joint work with Zhe Sun.
  A decorated surface S is an oriented topological surface with boundary, equipped with marked points on the boundary considered modulo the isotopy. 
  We consider the moduli space M(S) of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp,  equipped with  a horocycle. The  space M(S) carries a canonical volume form. However,  if the cusps are present,  the volume of the space M(S), as well as its   variant without horocycles,  are infinite. 
  We introduce the exponential  volume form, given by the volume form multiplied by the exponent of a canonical function on M(S).
  We show that  the exponential volume is finite. We prove  recursion formulas  for the exponential volumes,  generalising Mirzakhani's recursions for the volumes of  moduli spaces of hyperbolic surfaces.  
 We suggest that the moduli space M(S)  with the exponential volume form is the true analog of the moduli space M_{g,n}, relevant to the open string theory.