Geometric recursion

We propose a general theory whose main component are functorial assignments ∑→Ω_∑ ∈ E(∑), for a large class of functors E from a certain category of bordered surfaces (∑'s) to a suitable a target category of topological vector spaces. The construction is done by summing appropriate compositions of the initial data over all homotopy classes of successive excisions of embedded pair of pants. We provide sufficient conditions to guarantee these infinite sums converge and as a result, we can generate mapping class group invariant vectors Ω_∑ which we call amplitudes. The initial data encode the amplitude for pair of pants and tori with one boundary, as well as the "recursion kernels" used for glueing. We give this construction the name of "geometric recursion", abbreviated GR. As an illustration, we show how to apply our formalism to various spaces of continuous functions over Teichmueller spaces, as well as Poisson structures on the moduli space of flat connections. The theory has a wider scope than that and one expects that many functorial objects in low-dimensional geometry and topology should have a GR construction. The geometric recursion has various projections to topological recursion (TR) and we in particular show it retrieves all previous variants and applications of TR. We also show that, for any initial data for topological recursion, one can construct initial data for GR with values in Frobenius algebra-valued continuous functions on Teichmueller space, such that the ω_g,n of TR are obtained by integration of the GR amplitudes over the moduli space of bordered Riemann surfaces and Laplace transform with respect the boundary lengths. In this regard, the structure of the Mirzakhani-McShane identities served as a prototype of GR.