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

Big surfaces appear across different areas of mathematics, bringing together researchers from diverse mathematical backgrounds. In this talk, I will give an overview of some of the roles big surfaces play in various contexts and introduce problems and questions to consider.

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.

In this talk I will review basic facts on the Teichmüller space of finite-type surfaces using geodesic currents. Then, we will move to infinite-type surfaces and see via examples what are some of the issues and fixes in generalizing the theory. In particular we will show a result of Bonahon-Saric showing that we still have a Thurston boundary.

We provide a review of some standard properties of the Teichmüller distance on the Teichmüller spaces of infinite-type Riemann surfaces. Then, we link this to recent results on the trajectory structure of the horizontal foliations of finite-area holomorphic quadratic differentials on 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.

Infinite periodic translation surfaces sit somewhere between compact and more general infinite-type ones. Their structure allows us to explore aspects of infinite surfaces using some of the powerful tools from the well-developed theory of compact translation surfaces. In this talk, I will give a brief overview of what we know about the dynamics and geometry of these surfaces, using the wind-tree model, a periodic billiard model, as a guiding example. I will focus in particular on the problem of counting closed geodesics up to a given length.

In this talk, we will study the ergodicity of skew products on [0, 1) x Z (w.r.t. the product of Lebesgue and counting measure) where the base is an interval exchange transformation T on d intervals and the cocycle is given by the difference of characteristic functions of the intervals (0, 1/2) and (1/2, 1), which we refer to as the 1/2 cocycle. Such systems arise naturally as first-return maps of directional flows on certain infinite translation surfaces and have been extensively studied in the case d = 2 (i.e., when T is a rotation). By a classical result of Oren, for any irrational rotation T, the associated skew product (w.r.t. to the 1/2 cocycle) is ergodic. However, very little is known in the case d > 2. We show that for typical symmetric interval exchange transformations, the associated skew product (w.r.t. to the 1/2 cocycle) is ergodic. As a consequence, we can deduce the ergodicity of the vertical flow on certain infinite translation surfaces arising from hyperelliptic translation surfaces.

In this talk, we will introduce the notion of an end-periodic homeomorphism of an infinite-type surface. We will explore how the geometry of the associated mapping torus is related to certain topological and dynamical features of the end-periodic gluing map. In particular, we will see how the hyperbolic volume of the 3-manifold can be bounded in terms of a certain dynamical feature of the homeomorphism. This talk represents joint work with Autumn Kent, Heejoung Kim, Christopher Leininger, and Marissa Loving (in various configurations).

In this talk we will discuss some work on groups which connect self-similar and Higman-Thompson groups to big mapping class groups via "braiding". We will explain some results on the topological finiteness properties of the resulting groups, which are topological generalizations of the algebraic properties of being finitely generated and finitely presented.

We define and study the graph Houghton group, the subgroup of asymptotically rigid mapping classes of certain infinite graphs, focusing in particular on its finiteness properties. We also produce an explicit finite presentation for the group. While closely related to classical Houghton groups and their braided and surface variants, we show the graph Houghton group is not commensurable to any of these groups. This is joint work with Sanghoon Kwak, Brian Udall, and Jeremy West.

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.

Given a surface $\Sigma$, its fine curve graph, $\mathcal{C}^{\dagger}(\Sigma)$, is the graph whose vertices are essential simple closed curves in $\Sigma$ and edges connect curves that are disjoint. Long, Margalit, Pham, Verberne and Yao proved that the natural morphism $\mathrm{Homeo}(\Sigma)\longrightarrow\mathrm{Aut}(\mathcal{C}^{\dagger}(\Sigma))$ is an isomorphism of groups when $\Sigma$ is a closed orientable hyperbolic surface. Since $\mathrm{Homeo}{(\Sigma)}$ is naturally a topological group under the compact-open topology, we can ask if there is a natural topology on $\mathrm{Aut}(\mathcal{C}^{\dagger}(\Sigma))$ for which the natural embedding $\mathrm{Homeo}(\Sigma)\rightarrow\mathrm{Aut}(\mathcal{C}^{\dagger}(\Sigma))$ is topological. In work in progress with Roberta Shapiro, we answer this question in the affirmative by endowing $\mathrm{Aut}(\mathcal{C}^{\dagger}(\Sigma))$ with the pointwise convergence topology induced by the Vietoris topology on $\mathcal{C}^{\dagger}(\Sigma)$. We will also discuss the suitability of the Vietoris topology for $\mathcal{C}^{\dagger}(\Sigma)$ by proving that it is a Polish topology turning $\mathcal{C}^{\dagger}(\Sigma)$ into what is known as a Borel graph---thus providing a reasonable connection between the topology and the graph structure of the fine curve graph from a descriptive set theoretic point of view. This relation leads to some natural questions that we will discuss at the end of the talk.

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, I will introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface. The surface is of negative, but not necessarily finite, Euler characteristic, and emphasis is on the case in which the surface is of infinite type. The topology on this marked moduli space can be described in multiple ways, each reminiscent of Teichmüller space. The marked moduli space reduces to the usual Teichmüller space in case the surface is of finite type, but is quite distinct from the (quasiconformal) Teichmüller space in case the surface is of infinite type. Since a big mapping class group is a topological group with a nontrivial topology, a basic question is whether its action on the marked moduli space, by change of marking, is continuous. We answer this question in the affirmative. We also show that the marked moduli space is contractible.

The first part of this talk will be a quick introduction to the geometry and topology of big surfaces (surfaces with an infinitely generated fundamental group). Big surfaces include the Cantor tree, Loch Ness monster, and flute surfaces. Such surfaces admit a rich variety of hyperbolic geometric structures which are studied in a number of different ways. For surfaces with a finitely generated fundamental group (for example, compact surfaces) any two geometric structures on it are related by a quasiconformal (qc) homeomorphism. Such homeomorphisms control how distorted the geometry of the surface can become. As a result, the space of hyperbolic metrics on a compact surface (of genus bigger than 1) called the Teichmüller space is given by quasiconformal deformations. On the other hand, for a big surface, there is no natural choice of Teichmüller space as there are uncountably many. The second part of this talk will be on big mapping class groups. The mapping class group of a surface $\Sigma$ is its group of self-homeomorphisms modulo isotopy. If the fundamental group of $\Sigma$ is finitely generated, the mapping class group is finitely generated and acts faithfully on the Teichmüller space. On the other hand, for a big surface the mapping class group is infinitely generated (hence the name, big mapping class group) and does not act naturally on any Teichmüller space. The final part of this talk is on our classification of big mapping classes and a construction of various spaces of hyperbolic structures for which the big mapping class group acts faithfully. As an application of our work, we show that a big mapping class group is not algebraically isomorphic to the modular group (quasiconformal mapping class group) of any hyperbolic surface. This is joint work with Yassin Chandran.

Birman-Hilden theory studies the problem of relating the mapping class group of a surface to that of a cover. Since the early 1970s it has been studied for finite branched covering maps between surfaces of finite-type. In this talk I will present recent work addressing this problem, and some applications, in the context of branched covering maps, possibly of infinite degree, between topological surfaces that may be of infinite-type. This is joint work with Nestor Colin, Ruben Hidalgo, Israel Morales and Saul Quispe.

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.

We will discuss how to build topological spaces of infinite graphs and surfaces, so that the resulting topologies are Polish. This allows one to apply the tools of descriptive set theory. In particular, we will show that the Loch Ness monster surface is generic, and discuss the complexity of homeomorphism using the notion of Borel reducibility of equivalence relations. This is joint work with Jenna Zomback.