Related papers: WWPD elements of big mapping class groups
Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded…
An infinite-type surface $\Sigma$ is of type $\mathcal{S}$ if it has an isolated puncture $p$ and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface,…
We show that the extended based mapping class group of an infinite-type surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a…
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two…
Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod(S;p)$ on the…
We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $\Sigma$ be a connected, orientable surface of infinite type with…
We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve…
We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type…
We consider the class of countable groups possessing an action on a finite product of hyperbolic graphs where every infinite order element acts loxodromically. When the graphs are locally finite, we obtain strong structure theorems for the…
Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We show that the big mapping class group of a stable…
We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and the work of Mann--Rafi on the large-scale geometry of mapping class groups of…
We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of…
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…
Let $\varphi:V\times V\to W$ be a bilinear map of finite vector spaces $V$ and $W$ over a finite field $\mathbb{F}_q$. We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of…
We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping…
The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph…
We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the…
In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between…
We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…
We prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group.…