The main subject of the present, partially expository and partially original book are actions of some classes of classical groups such as free groups, surface groups and mapping class groups on the circle \(S^1\).
“The purpose of this monograph is to give a systematic construction of uncountable families of actions of these groups which have “essentially different” dynamics. The tools described allow us to construct many exotic actions of classically studied groups, i.e. actions which are not semi-conjugate to the “usual” or “standard” actions of these groups.”
Here two actions \(\rho_0, \rho_1\) in the non-linear representation variety \(\text{Hom}(L, \text{Homeo}_+(S^1))\) of homomorphisms of a finitely generated group \(L\) to the group of orientation-preserving homeomorphisms of \(S^1\) are semi-conjugate if there is a monotone degree-one map \(h: S^1 \to S^1\) such that \(h \circ \rho_0 = \rho_1 \circ h\). It is shown that, if \(L\) is a nontrivial limit group (including free groups and Fuchsian groups) then \(\text{Hom}(L, \text{Homeo}_+(S^1))\) contains uncountably many semi-conjugacy classes of faithful actions on \(S^1\) with pairwise disjoint rotation spectra (excluding 0) which lift to \(\mathbb R\). In particular, it is shown that for most Fuchsian groups this flexibility phenomenon occurs even locally (complementing a result of K. Mann [Invent. Math. 201, No. 2, 669–710 (2015; Zbl 1327.57002)]). Also, “we prove that each non-elementary free or surface group admits an action on \(S^1\) that is never semi-conjugate to any action that factors through a finite-dimensional connected Lie subgroup in \(\text{Homeo}_+(S^1)\)”.
Concerning mapping class groups, the authors prove that a mapping class group of a surface with non-empty boundary has at least two inequivalent actions on the circle, and hence in particular an action which is not equivalent to the standard Nielsen-type action (for closed surfaces, this would be excluded by a recent preprint of K. Mann and M. Wolff, [“Rigidity of mapping class group actions on \(S^1\)”, Preprint, arXiv:1808.02979]), and the authors ask if there are again uncountably many inequivalent actions.
The main tools of the book are a topological Baumslag Lemma (a criterion for certifying that elements of a free group do not reduce to the identity), coupled with a Baire category argument for representation varieties in order to prove a number of combination theorems for faithful but possibly indiscrete representations which are at the heart of the present text (in the spirit of the classical combination theorems for Kleinian groups).
The notion of semi-conjugacy goes back to É. Ghys [Enseign. Math. (2) 47, No. 3–4, 329–407 (2001; Zbl 1044.37033)] who exhibits a correspondence between cohomology classes and semi-conjugacy classes (“the bounded Euler class is a semi-conjugacy invariant”). Various related notions of semi-conjugacy have emerged which are summarized and shown to be equivalent in the present text (see also a paper by Bucher, Frigerio and Hartnick [M. Bucher et al., ibid. (2) 62, No. 3–4, 317–360 (2016; Zbl 1375.37123)]).
The book contains a lot of information and is written in a concise and well-organized way, describing its contents in a long introductory chapter and starting each chapter with a small abstract, with many references to the relevant literature and comments on related concepts and on related work (maybe sometimes the addition of more intuitive versions and motivations of some of the basic, sometimes quite technical definitions and concepts would have been helpful for a less experienced reader).