# RESEARCH

**Free products from spinning and rotating families **[arXiv] **- ***Submitted*

(Joint with M. Bestvina, R. Dickmann, G. Domat, P. Patel, and E. Stark)

**Free products from spinning and rotating families**[arXiv]

**-***Submitted*

(Joint with M. Bestvina, R. Dickmann, G. Domat, P. Patel, and E. Stark)

* 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

*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.