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    • Relatively Hyperbolic Extensions of Groups and Cannon-Thurston Maps

      Abhijit Pal

      Volume 120 Issue 1 February 2010 pp 57-68

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/120/01/0057-0068

    • Keywords

       

      Cannon–Thurston maps; relatively hyperbolic groups.

    • Abstract

       

      Let $1\to(K, K_1)\to(G, N_G(K_1))\to(\mathcal{Q}, \mathcal{Q}_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K_1$ a proper non-trivial subgroup of 𝐾 and 𝐾 strongly hyperbolic relative to $K_1$. Assuming that, for all $g\in G$, there exists $k_g\in K$ such that $gK_1g^{-1}=k_gK_1k^{-1}_g$, we will prove that there exists a quasi-isometric section $s:\mathcal{Q}\to G$. Further, we will prove that if 𝐺 is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon–Thurston map for the inclusion $i:\Gamma_K\to\Gamma_G$.

    • Author Affiliations

       

      Abhijit Pal1

      1. Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India

    • Dates

       
      Published
      February 2010
      Manuscript received
      17 September 2009
      Manuscript revised
      28 October 2009

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