Notes on Maskit's planarity theorem (Q2154801)

``In this paper we give a discussion of the Planarity Theorem of \textit{B. Maskit} [Ann. Math. (2) 81, 341--355 (1965; Zbl 0151.33003)], and some related results. This effectively gives a classification of groups of homeomorphisms of planar surfaces which have finite-type quotients. In particular, they can be described in terms of geometrically finite ``function groups'' acting as Möbius tranformations on the Riemann sphere. Topologically, they can be described as a class of orbifold fundamental groups of a particular class of 3-orbifolds, which generalize the standard notion of a compression body. They can also be described in terms of finitely generated groups with planar Cayley graphs, as described in a paper by \textit{A. Georgakopoulos} [Trans. Am. Math. Soc. 373, No. 7, 4649--4684 (2020; Zbl 1442.05086)] .'' ``We will also explain how one can give a different proof of the Planarity Theorem using the theory of tracks due to \textit{M. J. Dunwoody} [Invent. Math. 81, 449--457 (1985; Zbl 0572.20025)].'' Here a planar surface is a space homeomorphic to a connected open subset of the 2-sphere; the typical case is the ``Cantor surface'', the complement of a Cantor set in the 2-sphere. (For another description of the class of Kleinian function groups in terms of graphs of groups, see also a paper by \textit{M. Reni} [Proc. Lond. Math. Soc. (3) 67, No. 1, 200--224 (1993; Zbl 0840.30026)]).