Infinite-type loxodromic isometries of the relative arc graph

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 construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph $mathcal{A}(Sigma, p)$. J. Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other surfaces of type $mathcal{S}$. The elements we construct are the composition of three shift maps on $Sigma$, and we give an alternate characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of $Sigma$ and a standard shift map. We then explicitly find their limit points on the boundary of $mathcal{A}(Sigma,p)$ and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map$(Sigma,p)$ has an infinite-dimensional space of quasimorphisms.

Stable commutator length on big mapping class groups

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

Isometry groups of infinite-genus hyperbolic surfaces

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 infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

The first integral cohomology of pure mapping class groups

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 class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surfaces simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

Residual Finiteness Growths of Virtually Special Groups

Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $text{SL}_k(mathbb{Z})$, for $k > 2$, has residual finiteness growth $n^{k-1}$.