Introductory speaker: Marissa Loving (University of Wisconsin, Madison)
Title: Hyperbolization, the curve complex, and end-periodic homeomorphisms
Abstract: We will discuss mapping tori of pseudo-Anosov homeomorphisms, especially as they relate to Thurston's renowned hyperbolization theorem for closed 3-manifolds, before discussing a piece of Minsky's work towards the proof of the Ending Lamination Conjecture. Time permitting we will also introduce end-periodic homeomorphisms and their compactified mapping tori, making note of some of the striking similarities between psuedo-Anosov homeomorphisms in the finite-type setting and end-periodic homeomorphisms in the infinite-type setting.
Plenary speaker: Brandis Whitfield (Temple University)
Title: The geometry of end-periodic mapping tori
Abstract: One creates a fibered 3-manifold by thickening a surface by the interval and gluing its ends via a surface homeomorphism. In the finite-type setting, much is known about how the topological data of the gluing homeomorphism determine geometric information about the hyperbolic 3-manifold. Currently, there is a lot of research activity surrounding end-periodic homeomorphisms of infinite-type surfaces.
In this talk we'll discuss various recent works surrounding the topological and geometric structure of mapping tori of end-periodic homeomorphisms. Work of Field-Kim-Leininger-Loving show that under certain conditions, end-periodic mapping tori admit hyperbolic metrics. Landry-Minksy-Taylor give a natural embedding of the compactified end-periodic mapping tori into a closed fibered hyperbolic manifold. As an infinte-type analogy to work of Minsky in the finite-type setting, my work uses lamination data of the end-periodic homeormophism to detect short curves in the hyperbolic manifold.
Introductory speaker: Tarik Aougab (Haverford College)
Title: Matrices, actions, and surfaces
Abstract: We will discuss many different actions of 2x2 matrices, and how they interact. We'll introduce ideas of hyperbolic, euclidean, and affine geometry, and draw many pictures.
Plenary speaker: Paige Helms (University of Washington, Seattle)
Title: Reciprocal Foliations on Lattice Surfaces
Abstract: Sarnak defined a reciprocal geodesic on the modular surface H / SL(2, Z) to be one which passes through the image of the conjugacy class of forder 2 elements in SL(2, Z) and counted their growth according to length. We interpret SL(2, Z) as the stabilizer group of the action of SL(2, R) on the torus, and H / SL(2, Z) as the moduli space of tori. Erlandsson-Souto extended Sarnak's results to count reciprocal geodesics on H / G where G is a lattice. We apply these results to certain lattice surfaces, those which, under the action of SL(2, R), have a stabilizer group which is a lattice in SL(2, R); in particular, we examine Bouw-Möller surfaces and McMullen's family of genus 2 surfaces with stabilizer group not commensurable to a Triangle group. We also show that each reciprocal geodesic induces a pair of orthogonal foliations whose expansion and contraction factors are reciprocal.