Related papers: Three perfect mapping class groups
We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend…
In this note, we prove that the compactly supported mapping class group of a surface containing a genus $3$ subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order $6$…
We completely classify the orientable infinite-type surfaces $S$ such that $\operatorname{PMap}(S)$, the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of…
A normal subgroup of the (extended) mapping class group of a surface is said to be geometric if its automorphism group is the mapping class group. We prove that in the case of the Cantor tree surface, every normal subgroup is geometric. We…
We show that the pure mapping class group is uniformly perfect for a certain class of infinite type surfaces with noncompact boundary components. We then combine this result with recent work in the remaining cases to give a complete…
Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type…
We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in…
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for…
We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the…
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We…
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping…
We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping…
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 introduce several approaches to studying the Cantor-Bendixson decomposition of and the dynamics on the (topological) space of subgroups for various families of countable groups. In particular, we uncover the perfect kernel and the…
The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston…
We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the…
The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the…
The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed…
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses…
In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of…