Ghys and Sergiescu proved in the 80s that Thompson's group T , and hence F , admits actions by C ∞ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of C ∞ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of C 1 diffeomorphisms.Furthermore, we show that the group of Lodha-Moore has no nonabelian C 1 action on the interval. We also show that many Monod's groups H(A), for instance when A is such that PSL(2, A) contains a rational homothety x → p q x, do not admit a C 1 action on the interval. The obstruction comes from the existence of hyperbolic fixed points for C 1 actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable. 1 2 Some definitions and notation Definition 2.1. Let M be a manifold and Homeo(M ) the group of homeomorphisms of M . A subgroup G ⊂ Homeo(M ) is C r -smoothable (r ≥ 1) if it is conjugate in Homeo(M ) to a subgroup in Diff r (M ), the group of C r diffeomorphisms of M .Remark 2.2. Even if a certain subgroup G ⊂ Homeo(M ) is not C r -smoothable, it is still possible that the group G, as abstract group, admits C r actions on the manifold M .
This article is inspired by two milestones in the study of non‐minimal group actions on the circle: Duminy's theorem about the number of ends of semi‐exceptional leaves, and Ghys' freeness result in real‐analytic regularity. Our first result concerns groups of real‐analytic diffeomorphisms with infinitely many ends: if the action is non‐expanding, then the group is virtually free. The second result is a Duminy type theorem for minimal codimension‐one foliations: either non‐expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.
This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite graphs: these are the so-called hierarchical graphs. They possess a well-defined level structure and any level is built using a simple recursion. Stopping the construction at any finite level, we have a discrete random metric space when we set the edges to have random length (using a multiplicative cascade with fixed law m).We introduce a tool, the cut-off process, by means of which one finds that renormalizing the sequence of metrics by an exponential factor, they converge in law to a non-trivial metric on the limit space. Such limit law is stationary, in the sense that glueing together a certain number of copies of the random limit space, according to the combinatorics of the brick graph, the obtained random metric has the same law when rescaled by a random factor of law m. In other words, the stationary random metric is the solution of a distributional equation. When the measure m has continuous positive density on R + , the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation. We also investigate topological and geometric properties of the random space when m is log-normal, detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.