Related papers: Structure and Topological Generation of Big Mappin…
The extended mapping class group of a surface $\Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\Sigma$. We are able to show that the extended mapping class group of an…
In this note we prove that the mapping class group of a compact topological manifold $M$ with boundary is of finite type, under assumptions on its dimension and connectivity.
For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing…
We answer two questions about the topology of end spaces of infinite type surfaces and the action of the mapping class group that have appeared in the literature. First, we give examples of infinite type surfaces with end spaces that are…
We prove that, for any infinite-type surface $S$, the integral homology of the closure of the compactly-supported mapping class group $\overline{\mathrm{PMap}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive…
We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of…
The aim of this paper is to survey some aspects of mapping class groups with focus on their finite dimensional representations arising in topological quantum field theory.
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…
We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the…
We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.
We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander…
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space called the…
We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.
This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…
We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a…
In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all "separating twists", all "bounding pair maps", and all "commutators of simply…
We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $\Sigma$ on itself. Our main results are: (1) All conjugacy classes of $MCG(\Sigma)$ are meager for every…
We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We…
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these…
We give a criterion to prove that some groups are not acylindrically hyperbolic. As an application, we prove that the mapping class group of an infinite type surface is not acylindrically hyperbolic.