Swathi Krishna - Cannon-Thurston maps

Abstract

Given a Hyperbolic group and a hyperbolic subgroup, if the inclusion extends continuously to a map between their Gromov boundaries, the boundary map is called the Cannon–Thurston map. In this talk, we will first briefly look at the cases where the Cannon–Thurston map exists. Then we focus on this particular result: let $1\to N\to G\to Q\to 1$ be an Exakte Sequenz of non-elementary hyperbolic groups. By the work of Mahan Mj, we know that $N\to G$ admits a Cannon–Thurston map. Let $Q^{\prime }$ be quasi-isometrically embedded subgroup of Q. We will see that the pre-image $H$ of $Q^{\prime }$ in $G$ is hyperbolic and the inclusion $H\to G$ admits the Cannon–Thurston map.

This is based on a joint work with Pranab Sardar.

Talk

We define what a Hyperbolic metric space is. From now on, we will assume a proper, geodesic, hyperbolic metric space.

We define the geodesic boundary as the equivalence class of all geodesic which are a finite distance apart. Similarly we define the quasigeodesic boundary as an equivalence class of quasi-geodesics. The sequented boundary consists of diverging equivalence classes of sequences which are some “finite distance apart”

In “nice” setting those are homeomorphic and called the Gromov or Ideal boundary (Hyperbolic space).

Every quasi-isometric embedding may extend to a qi embedding of the boundary. This is a homeomorphism on the image. If the continuation to the boundary exists, it is called the Cannon–Thurston map.

Theorem

If $H\subseteq G$ is Quasi-convex, then $(H,G)$ admits a Cannon-Thurston map.

Definition (Connan-Thurston 1985, 2007)

Take a surface $S_g$ with $g\ge 2$. We look at the Mapping torus $M_{\varphi }$, defined by “some map” $\varphi$. The inclusion of the surface $S_g$ into $M_{\varphi }$ induces a map to the Universal covers ${\mathbb{H}}^2\to {\mathbb{H}}^3$. On the boundary this induced map is $\partial i:S^1\to S^2$ and continuous and surjective. This means it is a Raumfüllende Kurve.

This applies for all functions? Or just if $\varphi$ is Pseudo-Anosov Homeomorphism? Im slightly confused.

Theorem

The Cannon-Thurston map is not always existent. Here are some group where CT maps do exist.

1. Some Amalgamiertes Produkt
2. Kapovich-Sandas: Some double amalgamated products
3. “More examples in relatively hyperbolic groups”

It does not exist in group given by

• Baker-Riley 2013
• Matsuda-Oguna 2014
• Haldu-Mj-Sardan 2025
• Benjamin-Coa-Gardam-Gupta-Stark 2021 (CAT(0) graphs)
• Charney-Cordan-Goldeborough-Sato-Zindau

I expected the talk to be much more fun…