Structure and Regularity of Group Actions on One-Manifolds

Given a smooth, orientable manifold \( M \) and a positive integer \( k \), \( \mathrm{Diff}_{+}^{k}(M) \) is defined to be the group of orientation-preserving homeomorphisms \( f : M \rightarrow M \) such that the derivatives of \( f \) and \( f^{-1} \) up to order \( k \) exist and are continuous. For \( \tau \in [0,1) \),   \( \mathrm{Diff}_{+}^{k, \tau}(M) \) is defined to be the group of functions in \( \mathrm{Diff}_{+}^{k}(M) \) whose \( k \)’th order derivatives are \( \tau \)-- Hölder continuous. This book studies the groups \( \mathrm{Diff}_{+}^{k, \tau}(M) \) and their finitely generated subgroups when \( M \) is a (usually compact) one-manifold such as the circle \( S^{1} \) or the closed interval \( I = [0, 1] \). The main focus of the text is on the interplay between the algebraic structure of groups \( G \leq \mathrm{Diff}_{*}^{k, \tau} (M) \), the regularity of the action (i.e., the values of \( k \) and \( \tau \)), and the arrangement of G-orbits of points in \( M \). This interplay combines ideas from group theory, analysis, and dynamics.

 

The book starts with a detailed survey of classical and recent results on diffeomorphism groups and smoothability of group actions with an eye towards developing the machinery to find embeddings of groups into \( \mathrm{Diff}_{+}^{k, \tau}(M) \) or to prove that no such embeddings exist. This machinery is then applied to a variety of classes of groups including nilpotent groups, solvable Baumslag-Solitar groups, chain groups such as Thompson’s group \( F \), right-angled Artin groups, and finite-index subgroups of mapping class groups of surfaces.  In most of these situations, a group G either admits an embedding into \( \mathrm{Diff}_{+}^{\infty}(M) := \cap_{k \geq 1} \mathrm{Diff}_{+}^{k}(M) \) or will embed into \( \mathrm{Diff}_{+}^{1}(M) \) but not \( \mathrm{Diff}_{+}^{2}(M) \). In contrast, the book culminates in a presentation of the authors’ theorem that for any \( k \) and \( \tau \) as above, there exists a finitely generated group \( G_{k,\tau} \) which embeds into \( \mathrm{Diff}_{+}^{k, \tau}(M) \) but does not embed into \( \mathrm{Diff}_{+}^{l,\epsilon}(M) \)  whenever \( k+ \tau <  l + \epsilon \) and \( M \in \left\{ S^{1},I \right\} \).

 

This book assumes only basic familiarity with groups, topology, and analysis on the level which can be found in introductory graduate courses or advanced undergraduate courses. As such, it should be suitable for most researchers and graduate students with an interest in learning about differentiable group actions. The authors give complete proofs of all of the main results that appear while also carefully explaining how each technical step fits into the overall strategy of each proof. They often state main results at the beginning of a section, followed by an outline of the main ideas in the argument, and finally give detailed proofs for each step. This structure is mirrored in the book itself as the first chapter provides an overview of the main  theorems covered and many of the main ideas behind these theorems with only minimal technical details.

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Michael Hull is an assistant professor at the University of North Carolina-Greensboro.  His interests include geometric group theory and topology.