This week's seminars
Pure Maths Colloquium: Quasispheres and Expanding Thurston maps
11 February 2019 16:00 in CM219
A quasisymmetric map is one that changes angles in a controlled way. As such they are generalizations of conformal maps and appear naturally in many areas, including Complex Analysis and Geometric group theory. A quasisphere is a metric sphere that is quasisymmetrically equivalent to the standard $2$-sphere. An important open question is to give a characterization of quasispheres. This is closely related to Cannon's conjecture. This conjecture may be formulated as stipulating that a group that ``behaves topologically'' as a Kleinian group ``is geometrically'' such a group. Equivalently, it stipulates that the ``boundary at infinity'' of such groups is a quasisphere.
A Thurston map is a map that behaves ``topologically'' as a rational map, i.e., a branched covering of the $2$-sphere that is postcritically finite. A question that is analog to Cannon's conjecture is whether a Thurston map ``is'' a rational map. This is answered by Thurston's classification of rational maps.
For Thurston maps that are expanding in a suitable sense, we may define ``visual metrics''. The map then is (topologically conjugate) to a rational map if and only if the sphere equipped with such a metric is a quasisphere. This talk is based on joint
work with Mario Bonk.
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