arXiv research

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…

Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.

Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.

This thesis introduces big mapping class groups and their structure.

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

Classifies big mapping classes on infinite type surfaces.

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-…

The study examines when mapping class groups are quasi-isometric to graphs of curves.

Paper shows mapping class groups are not extremely amenable except for specific cases.

We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of …

Overview of infinite surface mapping class groups.

Proves left-orderability of mapping class groups of infinite-type surfaces.

Study on self-similar surfaces and their mapping class groups generated by involutions.

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 `big' mapping class groups and shows that each finite-index subgroup has finite ou…

Multitwists cannot generate all compactly supported mapping class groups on infinite-type surfaces.

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.

By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.

Involutions generate mapping class groups of infinite surfaces.

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 parts of the paper are devoted to the definition of objects and tools needed to int…

Let $Γ$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $Γ$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.

Study shows mapping class groups are one-ended for surfaces with at least one end.

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

Parabolic mapping class acts on curve graphs of infinite type surfaces.

Study of CB generating sets for infinite-type surfaces.

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

Study on mapping class groups of infinite type surfaces.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

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 surfaces that determines in many cases whether there is such an action. This allows us …

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(Λ^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(Λ^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorn…

Study on compact and finite-type support in mapping class group homology.

Study the relationship between orbit braid group and equivariant mapping class group on surfaces.

Proves homology of mapping class groups for infinite-type surfaces.

Study of conjugacy classes in infinite-type surfaces' mapping class groups.

Study large-scale geometry of infinite type surface mapping class groups.

Study of infinite type surfaces' mapping class groups via hyperbolic structures.

The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.

Study shows compact mapping class groups of infinite type surfaces are never perfect.

A metric space $M$ us said to have the fibered approximation property in dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $ε>0$, $m\geq 0$ and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$ such that $g'$ is $ε$-homotopic to $g$ and $\dim g'\big(\{z\}\times I^n\big)\leq n$ for all $z\i…

The paper explores the twisted Rokhlin property in mapping class groups of surfaces.

The paper proves conditions for curvature blow-up in quiescent big bang singularities.

Maps on certain manifolds decrease area and scalar curvature is bounded.

Study shows surfaces without certain curves have infinite orbit graph.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer $n \geq 2$. In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT represen…

Study of flip graphs and their automorphism groups for infinite-type surfaces.

Surface Houghton groups are studied for their mapping class properties.

Uniform volume estimate for Kähler metrics in big cohomology classes.