# RESEARCH

## Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps [arXiv]

(Joint with Ryan Dickmann and Hannah Hoganson)

(Joint with Ryan Dickmann and Hannah Hoganson)

Abstract : Motivated by the recent work of Algom-Kfir and Bestinva introducing the mapping class group of an infinite graph via proper homotopy equivalences, we give a necessary and sufficient condition for a surface to be properly homotopy equivalent to a graph. We consider second-countable orientable surfaces that are possibly infinite-type and have noncompact boundary. For surfaces proper homotopy equivalent to graphs, we explore the basic properties of the induced map between the mapping class groups of the surface and the graph. We view this induced map as the basis of a Dehn-Nielsen-Baer analog in the setting of infinite-type surfaces.

## Nonunique Ergodicity on the Boundary of Outer space [arXiv]

(Joint with Mladen Bestvina and Elizabeth Field)

(Joint with Mladen Bestvina and Elizabeth Field)

Abstract : To an ℝ-tree in the boundary of Outer space, we associate two simplices: the simplex of projective length measures, and the simplex of projective dual currents. For both kinds of simplices, we estimate the dimension of maximal simplices for arational ℝ-trees in the boundary of Outer space.

## Coarsely Bounded Generating Sets for Mapping Class Group of Infinite-type Surfaces [arXiv]

(Joint with Thomas Hill, and Rebecca Rechkin)

(Joint with Thomas Hill, and Rebecca Rechkin)

Abstract : Mann-Rafi's work takes a first step toward studying the coarse geometry of the mapping class group of an infinite-type surface. They accomplish this by constructing a coarsely bounded (CB) generating set for the mapping class groups of such surfaces. In this expository note, we illustrate the generating sets for various explicit examples.

## Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs [arXiv]

(Joint with George Domat, and Hannah Hoganson)

(Joint with George Domat, and Hannah Hoganson)

Abstract : We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.

Abstract : In his paper, Thurston shows that a positive real number h is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant λ=eʰ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof is difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.

Abstract : We prove that the full automorphism group and the outer automorphism group of the free group of countably infinite rank are coarsely bounded. That is, these groups admit no continuous actions on a metric space with unbounded orbits, and have the quasi-isometry type of a point.

Abstract : We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and the work of Mann--Rafi on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact).

Our techniques also give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all of the locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.

Abstract : The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group G. Their work gives conditions under which the normal closure in G is a free product. In this paper we unify their results and simplify and significantly shorten the proof of Dahmani-Guirardel-Osin.

Abstract : We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold:

(a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words,

(b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmüller space, and

(c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length.

## Slides

### Lighting Talk @ 2019 Tech Topology Conference, Georgia Institute of Technology, USA

### Lab/Research Seminar

(03/22/2018)Curve complex of Torus

(04/26/2018)Teichmüller space of Torus

(05/10/2018)Hyperbolicity of curve complex

(05/31/2018)Translation length on curve graph

(08/23/2018)Distances in curve graph

(10/02/2018)[Masur-Minsky Series]I. Introduction and main results

(10/31/2018)[Masur-Minsky Series]II. Outline of the proof of hyperbolicity

### MAS532, Algebraic Topology II, KAIST

(10/29/2018)Homotopy theory of fiber bundles

## Posters

*Poster template is courtesy of Mike Morrison #BetterPoster, and its LaTeX version is due to Rafael Balio.

*The QR code for the preprint is no longer available.

*Poster template courtesy of Mike Morrison #BetterPoster, and its LaTeX version is due to Rafael Balio.