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Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ S2. We prove that G admits such an action that is in addition co-compact, provided we can replace X by another surface Y ⊆ S2. We also prove that if a group H has a finitely generated Cayley (multi-) graph C equivariantly embeddable in S2, then C can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere. [ABSTRACT FROM AUTHOR]

We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question is whether each equilibrium configuration needs to be an arithmetic progression. Under very broad assumptions on the force we show this for the particles on the circle. In the case of infinitely many particles on the line we show the same result under the assumption that the maximal (or the minimal) gap between successive points is finite (positive) and assumed at some pair of successive points. Under the assumption of analyticity for the force field (e.g. the Coulomb force) we deduce some extra rigidity for the configuration: knowing an equilibrium configuration of points in a half-line determines it throughout. Various properties of the equlibrium configuration are proved. [ABSTRACT FROM AUTHOR]