• Abhijit Pal

      Articles written in Proceedings – Mathematical Sciences

    • Relatively Hyperbolic Extensions of Groups and Cannon-Thurston Maps

      Abhijit Pal

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      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 $i:\Gamma_K\to\Gamma_G$.

    • A Note on Stable Teichmüller Quasigeodesics

      Abhijit Pal

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      In this note, we prove that for a cobounded, Lipschitz path $\gamma:I\to \mathcal{T}$ in the Teichmüller space $\mathcal{T}$ of a hyperbolic surface, if the pull back bundle $\mathcal{H}_\gamma\to I$ of the cannonical $\mathbb{H}^2$-bundle $\mathcal{H}\to T$ is a strongly relatively hyperbolic metric space then there exists a geodesic 𝜉 of 𝑇 such that $\gamma(I)$ and 𝜉 are close to each other.

    • Strongly contracting geodesics in a tree of spaces


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      Let $X$ be a tree of proper geodesic spaces with edge spaces strongly contracting and uniformly separated from each other by a number depending on the contraction function of edge spaces. Then we prove that the strongly contracting geodesics in vertex spaces are quasiconvex in $X$. We further prove that in $X$ if all the vertex spaces are uniformly hyperbolic metric spaces, then $X$ is a hyperbolic metricspace and vertex spaces are quasiconvex in $X$.

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