arXiv research

We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

Study of point-pushing actions on manifolds with boundary.

A filling curve $γ$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(γ)$ of $S$ via the process of "point-pushing along $γ$." We consider the relationship between the self-intersection number $i(γ)$ of $γ$ and the dilatation of $P(γ)$; our main result is that the dilatation is bounded between $(i(γ)+1…

We study the Birman exact sequence for compact 3-manifolds.

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute low…

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As…

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group…

Given $φ$ a pseudo-Anosov map, let $\ell_\mathcal{T}(φ)$ denote the translation length of $φ$ in the Teichmüller space, and let $\ell_\mathcal{C}(φ)$ denote the stable translation length of $φ$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $χ(S)$, the minimal po…

Study moduli space of quadratic differentials with new geometric insights.

We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 …

Random curves on surfaces have predictable properties as they grow.