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We consider the space 
$\mathcal{P}_d$ of smooth complex projective plane curves of degree 
$d$. There is the tautological family of plane curves defined over 
$\mathcal{P}_d$, which has an associated monodromy representation 
$\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ into the mapping class group of the fiber. For 
$d \le 4$, classical algebraic geometry implies the surjectivity of 
$\rho _d$. For 
$d \ge 5$, the existence of a 
$(d-3)^{rd}$ root of the canonical bundle implies that 
$\rho _d$ cannot be surjective. The main result of this paper is that for 
$d = 5$, the image of 
$\rho _5$ is as large as possible, subject to this constraint. This requires combining the algebro-geometric work of Lönne with Johnson’s theory of the Torelli subgroup of 
$\textrm{Mod}(\Sigma _g)$.
Let 
$M$ be an oriented smooth manifold and 
$\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of 
$M$, where 
$\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group 
$\operatorname{Homeo}_0\!(M,\omega )$ and 
$\operatorname{Homeo}_+\!(M,\omega )$, respectively, and in several cases prove their non-triviality. More precisely, we define:
• Volume classes in 
$\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$, where 
$M$ is a hyperbolic manifold of dimension 
$n$.
• Euler classes in 
$\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$, where 
$S$ is an oriented closed hyperbolic surface.
We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic 
$3$-manifolds; hence, they are non-trivial.
The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over 
$\mathbb {C}(\!(t)\!)$ and show that the Ceresa class in each of these settings is torsion.
We study the
$E_2$
-algebra
$\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$
consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion
$\Omega B\Lambda \mathfrak {M}_{*,1}$
: it is the product of
$\Omega ^{\infty }\mathbf {MTSO}(2)$
with a certain free
$\Omega ^{\infty }$
-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups
$\Gamma _{g,n}$
with
$g\geqslant 0$
and
$n\geqslant 1$
.
We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level 
$m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.
The germ of the universal isomonodromic deformation of a logarithmic connection on a stable 
$n$-pointed genus 
$g$ curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus 
$g>0$, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.
The group of
${\mathcal{C}}^{1}$
-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations
$nV$
of Thompson’s group
$V$
arise when we consider products of central ternary Cantor sets. We derive that the
${\mathcal{C}}^{2}$
-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.
