Understanding the Use of Cannon Thurston Cards | by Monodeep Mukherjee | August 2022

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  1. Research Announcement: A combination theorem for acylindrical complexes of hyperbolic groups and Cannon-Thurston maps(arXiv)

Author : Pranab Sardar, Ravi Tomar

Summary : This is an announcement of some of the results obtained as part of the second author’s doctorate. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is hyperbolic in which the vertex groups are quasi-convex. In the second part of the article, we prove the existence of Cannon-Thurston maps for certain subcomplexes of groups in acylindrical complexes of hyperbolic groups (see Theorem 0.4)

2. Cohomology fractals, Cannon-Thurston maps and geodesic flux(arXiv)

Author : David Bachman, Matthias Goerner, Saul Schleimer, Henry Segerman

Summary : Cohomology fractals are images naturally associated with cohomology classes in the three hyperbolic manifolds. We generate these images for the three cusped, incomplete and closed hyperbolic manifolds in real time by ray tracing up to a fixed visual radius. We discovered cohomology fractals by trying to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing images of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behavior of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values ​​of cohomology fractals are normally distributed, but with divergent standard deviations. In fact, cohomology fractals do not converge to a limit function. Instead, we show that the limit is a distribution on the sphere at infinity, depending only on the manifold and the cohomology class

3. Cannon-Thurston (arXiv) Metric Beam and Map Pullbacks

Author : Wathi Krishna, Pranab Sardar

Summary : We introduce the notion of metric bundle pullbacks. Given a metric bundle (graph) X on B where X and all fibers are uniformly (Gromov) hyperbolic and non-elementary, and a Lipschitz qi plunging i:A→B we show that the pullback i∗X is hyperbolic and l he map i∗:i∗X→X admits a continuous boundary extension, ie a Cannon-Thurston (CT) map ∂i∗:∂(i∗X)→∂X. As an application of our theorem, we show that, given a short exact sequence of non-elementary hyperbolic groups 1→N→G→πQ→1 and a buried subgroup qi of finite type Q1

4. Cannon-Thurston Maps(arXiv)

Author : Mahan Mj

Summary : We give an overview of the theory of Cannon-Thurston maps which constitutes one of the links between the analytic and hyperbolic complex geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.

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