Groups acting on the circle with invariant laminations

Abstract

By Thurston and Calegari-Dunfield, it was shown that any fundamental group of an atoroidal 3-manifold with taut foliation acts on a circle, so-called a universal circle, by orientation preserving homeomorphisms. From the proof, we may observe that the universal circle actions preserve pairs of laminations of S^1 . In this paper, we study the algebraic or dynamic properties of such group actions in a general sense. A laminar group is a subgroup of the orientation preserving circle homeomorphism group Homeo+(S^1), preserving a lamination of S^1 . In this paper, we show a kind of Tits alternative for tight pairs which are laminar groups with some extra conditions. We then show that any laminar group preserving three invariant laminations with some transversality condition is a convergence group and so it is a discrete Mo ̈bius subgroup of Homeo+(S^1) by the convergence group theorem. Conversely, we also show that every Fuchsian group G of the first kind such that H/G is not a geometric pair of pants is such a laminar group.