We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by C 1`bv diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by C 1`bv diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length three. As a corollary, we also show that no finite index subgroup of AutpF n q and OutpF n q for n ě 3, the Torelli group for genus at least 3, and of each term of the Johnson filtration for genus at least 5, can act faithfully by C 1`bv diffeomorphisms on a compact one-manifold.
Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.
We propose a program to study groups acting faithfully on S 1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S 1 is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3-manifolds developed by Thurston and Calegari-Dunfield.
We give a complete description of the closure of the space of one-generator closed subgroups of PSL 2 (R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) ∼ = PSL 2 (R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps ϕ n : X → Y transform convergent sequences of closed subsets of the domain X into convergent sequences of closed subsets of the range Y . In particular, this allows us to compute certain geometric limits of PSL 2 (R) only by looking at the Hausdorff limit of some closed subsets of C.
We describe the topology of the space of all geometric limits of closed abelian subgroups of PSL 2 (C). Main tools and ideas will come from the previous paper [BC12]. * We really appreciate that John H. Hubbard let us know about this problem and explained how we could approach at the beginning. He also has provided us a lot of advices through enlightening discussions. We also thank to Bill Thurston for the helpful discusstions.
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