Schedule for: 25w5359 - Blooming Beasts: A Conference on the Geometry, Topology, and Dynamics of Infinite-Type Surfaces

The Nielsen-Thurston classification for mapping classes on finite-type surfaces provides a canonical way to divide the surface into subsurfaces such that on each piece, the mapping class is either periodic or pseudo-Anosov. In this talk, I will explain why this exact statement cannot hold for infinite-type surfaces, and describe some recent work of Bestvina, Fanoni, and Tao towards a generalization of this classification for mapping classes that don’t display any pseudo-Anosov-like behavior. At the other end of the spectrum, there are several possible approaches for defining an analogue of a pseudo-Anosov mapping class. I will discuss one such approach: classifying the mapping classes that stabilize two transverse laminations.

Building on the background provided by Carolyn Abbott’s talk, I will discuss two approaches to the problem of understanding the generalization of pseudo-Anosov maps for infinite-type surfaces. The first approach is to construct and classify those mappings classes that act loxodromically on a curve and arc graph associated to the surface. The second approach is to classify those mapping classes which give rise to hyperbolic mapping tori.

Compact translation surfaces have a very rich theory, which mixes geometry, topology and dynamics. On the other hand, translation surfaces of infinite type have a much less developed theory, and much is still unknown. We will give a short introduction, of the geometry and dynamics of infinite translation surfaces, focusing on examples and open problems.

Compact translation surfaces have a very rich theory, which mixes geometry, topology and dynamics. On the other hand, translation surfaces of infinite type have a much less developed theory, and much is still unknown. We will give a short introduction, of the geometry and dynamics of infinite translation surfaces, focusing on examples and open problems.

Endperiodic maps are a class of homeomorphisms of infinite-type surfaces whose compactified mapping tori have a natural depth-one foliation. By work of Landry-Minsky-Taylor, every atoroidal endperiodic map is homotopic to a type of map called a spun pseudo-Anosov. Spun pseudo-Anosovs share certain dynamical features with the more familiar pseudo-Anosov maps on finite-type surfaces. A theorem of Thurston states that pseudo-Anosovs minimize the number of periodic points of any given period among all maps in their homotopy class. We prove a similar result for spun pseudo-Anosovs, strengthening a result of Landry-Minsky-Taylor.

A topological group is amenable if its every continuous action on a compact set admits an invariant probability measure. The Tits alternative is a result about the amenable/non-amenable dichotomy of subgroups of a given group, in which all non-amenable subgroups contain a non-abelian free subgroup. This property is first proven by Tits (1972) for linear groups and it soon becomes one of the central themes in geometric group theory. Same results hold for mapping class group of finite type surfaces, shown by McCarthy (1985) and Ivanov (1986), as well as for Out(F_n), by Bestvina and Handel (2000). But the mapping class groups of infinite-type surfaces and graphs fail to have this alternative. So a natural question: when can a closed subgroup of such a group be amenable? We show that these amenable groups have to be nowhere dense. Moreover, if the graph is a tree, then we also show that its mapping class group is amenable under certain conditions.

Given two infinite type surfaces $\Sigma$ and $\Sigma'$, one can ask whether they share a common finite-sheeted cover. In this case we say that $\Sigma$ and $\Sigma'$ are commensurable, and under this equivalence relation we can form the commensurability class of a given surface. It is natural to ask if all commensurability classes contain a minimal element - a surface that is covered by all others in its class. This talk will cover results from joint work with Kasra Rafi addressing the existence of such elements for surfaces with countable, stable, end spaces. In particular, we show that there exists a surface with countable, stable end space whose commensurability class does not contain a minimal element, disproving the theorem in full generality. However, under certain stronger hypotheses on the end space of $\Sigma$, we are able to prove the existence of minimal elements for some surfaces of this type.

In this talk we introduce a Denjoy homeomorphisms of the circle, which can be thought as "irrational rotations of the circle that leave an invariant Cantor set". Using these homeomorphisms we create uncountably-many homeomorphisms of the Cantor-tree surface, all of which have mapping torus homeomorphic to (the interior of) a genus 2 handlebody. Finally, we show some variations of this construction to a) fiber any genus g > 1 handlebody with Cantor-tree surfaces, and b) how can we modify the construction of the homeomorphisms for this result to be applied to different surfaces. This is joint work with Christopher J. Leininger and Ferrán Valdez.