arXiv subjects
Publications and source records attributed to Autumn E. Kent.
We show that there is a type-preserving homomorphism from the fundamental group of the figure-eight knot complement to the mapping class group of the thrice-punctured torus. As a corollary, we obtain infinitely many commensurability classes of purely pseudo-Anosov surface subgroups of mapping class groups of closed surfaces. This gives the first examples of compact atoroidal surface bundles over surfaces.
For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of ${\rm MCG}(Σ)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(Σ)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of ${\mathcal I}(Σ)$ coincides with ${\rm MCG}(Σ)$. This extends the results for finite-type surfaces of Farb-Ivanov, Brendle-Margalit and KIda to the setting of infinite-type surfaces.
We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any $ε>0$, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least $1-ε$. As R approaches infinity, and as $ε$ approaches zero, this gives a sequence of knots that is said to Benjamini--Schramm converge to hyperbolic space. This answers a question of Brock and Dunfield.