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Vaughan Jones discovered unexpected connections between Richard Thompson'sgroup and subfactor theory while attempting to construct conformal fieldtheories (in short CFT). Among other this founded Jones' technology: a powerfulnew method for constructing actions of fraction groups which had numerousapplications in mathematical physics, operator algebras, group theory and moresurprisingly in knot theory and noncommutative probability theory.  We propose and outline a program in the vein of Jones' work but where theThompson group is replaced by a family of groups that we name forest-skeingroups. These groups are constructed from diagrammatic categories, aretailor-made for using Jones' technology, capture key aspects of the Thompsongroup, and aim to better connect subfactors with CFT. Our program strengthensJones' visionary work and moreover produces a plethora of concrete groups whichsatisfy exceptional properties.  In this first article we introduce the general theory of forest-skein groups,provide criteria of existence, give explicit presentations, prove that theirfirst L$^2$-Betti number vanishes, construct a canonical action on a totallyordered set, establish a topological finiteness theorem showing that many ofour groups are of type $F_\infty$, and finish by studying a beautiful class ofexplicit examples.

Forest-skein groups I: between Vaughan Jones' subfactors and Richard Thompson's groups