#### Date of Submission

8-28-2009

#### Date of Award

8-28-2010

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Mj, Mahan (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Let P : Y â†’ T be a tree of strongly relatively hyperbolic spaces such that Y is also a strongly relatively hyperbolic space. Let X be a vertex space and i : X Ö’â†’ Y denote the inclusion. The main aim of this thesis is to extend i to a continuous map i : X â†’ Y , where X and Y are the Gromov compactifications of X and Y respectively. Such continuous extensions are called Cannon-Thurston maps. This is a generalization of [Mit98b] which proves the existence of Cannon-Thurston maps for X and Y hyperbolic. By generalizing a result of Mosher [Mos96], we will also prove the existence of a Cannon-Thurston map for the inclusion of a strongly relatively hyperbolic normal subgroup into a strongly relatively hyperbolic group. Let us first briefly sketch the genesis of this problem.Let H be an infinite quasi-convex subgroup of a word hyperbolic group G. We choose a finite generating set of G that contains a finite generating set of H. Let Î“H, Î“G be their respective Cayley graphs with respect to these finite generating sets. Let âˆ‚Î“H and âˆ‚Î“G be hyperbolic boundaries of Î“H and Î“G respectively. Then it is easy to show that the inclusion i: Î“H â†’ Î“G canonically extends to a continuous map from Î“H âˆª âˆ‚Î“H to Î“G âˆª âˆ‚Î“G. But if H is not quasi-convex, it is not clear whether there is such an extension. It turns out that for a wide class of non-quasiconvex subgroups such an extension is possible. The first example of this sort was given by J.Cannon and W.Thurston in [CT07] (1989). They showed that if G is the fundamental group of a closed hyperbolic 3-manifold M fibering over a circle with fiber a closed surface S and if H is the fundamental group of S, then there exists a continuous extension for the embedding i: Î“H â†’ Î“G. In [Min94], Y.N.Minsky generalized Cannon-Thurstonâ€™s result to bounded geometry surface Kleinian groups without parabolics. Later on, Mitra, in [Mit98a, Mit98b] (1998), gave a different proof of Cannon-Thurstonâ€™s original result and generalized it in the following two directions:Theorem 0.0.1. (Mitra [Mit98a]) Let G be a hyperbolic group and let H be a hyperbolic subgroup that is normal in G. Let i: Î“H â†’ Î“G denote the inclusion. Then i extends to a continuous map Ëœi: Î“H âˆª âˆ‚Î“H â†’ Î“G âˆª âˆ‚Î“G.Theorem 0.0.2. (Mitra [Mit98b]) Let (X, d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T. If X is hyperbolic then there exists a Cannon-Thurston map for i: Xv â†’ X, where Xv is the vertex space corresponding to v.Let Î£ be a compact surface of genus g(Î£) â‰¥ 1 with a finite non-empty collection of boundary components {C1, ..., Cm}. Subgroups of Ï€1(Î£) corresponding to the fundamental groups of the boundary curves are called peripheral subgroups. Consider a discrete and faithful action of Ï€1(Î£) on H3 . The action is strictly type preserving if the maximal parabolic subgroups are precisely the peripheral subgroups of Ï€1(Î£). Let N be the quotient manifold obtained from H3 under this action.

#### Control Number

ISILib-TH301

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Pal, Abhijit Dr., "Cannon-Thurston Maps and Relative Hyperbolicity." (2010). *Doctoral Theses*. 28.

https://digitalcommons.isical.ac.in/doctoral-theses/28

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842804