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Register a new userVirtual critical regularity of mapping class group actions on the circle
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We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,tau}$ action of $(G_1times G_2)*mathbb{Z}$ on $S^1$ is faithful for $tau>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.
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In this monograph, we give an account of the relationship between the algebraic structure of finitely generated and countable groups and the regularity with which they act on manifolds. We concentrate on the case of one--dimensional manifolds, culminating with a uniform construction of finitely generated groups acting with prescribed regularity on the compact interval and on the circle. We develop the theory of dynamical obstructions to smoothness, beginning with classical results of Denjoy, to more recent results of Kopell, and to modern results such as the $abt$--Lemma. We give a classification of the right-angled Artin groups that have finite critical regularity and discuss their exact critical regularities in many cases, and we compute the virtual critical regularity of most mapping class groups of orientable surfaces.
We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points w_k and a sequence of radii r_k so that the ball B(w_k, r_k) is disjoint from a quasi-axis a of f, but for any projection map from mapping class group to a, the diameter of the image of B(w_k, r_k) grows like log(r_k).
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $partial X$. We define the critical exponent $delta(mu)$ of any discrete invariant random subgroup $mu$ of the locally compact group $G$ and show that $delta(mu) > frac{d}{2}$ in general and that $delta(mu) = d$ if $mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdans property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.
We give a short proof of Masbaum and Reids result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.
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