We prove various results that, given a sufficiently rich subgroup G of the group of homeomorphisms on the real line, describe the structure of the other possible actions of G on the line, and address under which conditions such actions must be semi-conjugate to the natural defining action of G. The main assumption is that G should be locally moving, meaning that for every open interval the subgroup of elements fixing pointwise its complement, acts on it without fixed points. One example (among many others) is given by Thompson's group F. In Part I, we show that when G is a locally moving group, every C1 action of G on the real line without fixed points is semi-conjugate to its standard action or to a non-faithful action. It turns out that the situation is much wilder when considering actions by homeomorphisms: for a large class of groups, including Thompson's group F, we describe uncountably many conjugacy classes of faithful minimal actions by homeomorphisms on the real line. In Part I I, we prove structure theorems describing the dynamics of exotic C0 actions, based on the study of laminar actions, which are actions on the line preserving a lamination. When G is a group of homeomorphisms of the line acting minimally, and with a non-trivial compactly supported element, then any faithful minimal action of G on the line is either laminar or conjugate to its standard action. Moreover, when G is a locally moving group satisfying a suitable finite generation condition, we prove that for any faithful minimal laminar action on the line, there is a map from the lamination to the line, called a horograding, which is equivariant with respect to the action on the lamination and the standard action, and satisfies some extra suitable conditions. This establishes a tight relation between all minimal actions on the line of such groups, and their standard actions. Among the various applications of this result, we show in Part III that for a large class of locally moving groups, the standard action is locally rigid, in the sense that every sufficiently small perturbation in the compact-open topology gives a semi-conjugate action. This is based on an analysis of the space of harmonic actions on the line for such groups. Along the way we introduce and study several concrete examples.
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.
We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of $L^2$ harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that $L^2$ harmonic forms decay faster at infinity than an arbitrary $L^2$ form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of $L^2$ harmonic forms is used in a companion paper to study the $L^2$ cohomology of some $QFB$ metrics.
