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October 21st 2012

3d hyperbolic limit sets

Vladimir Bulatov released a few very nice videos of 3d limit sets of hyperbolic group actions. Two of them are displayed below, but be sure to check his youtube channel for more.

And here are some explanations about what I understand of what these sets are. The hyperbolic space in 4 dimensions has a "Poincaré ball model", analogous to the Poincaré disk model in 2 dimensions, in which it is represented by a 4-dimensional ball. The boundary of the 4-dimensional ball is a 3-dimensional sphere. Now consider a discrete group of isometries of the 4-dimensional hyperbolic space and the associated orbit of a point. Points in the orbit might accumulate near certain points of the 3-sphere. The points for which this occur form the limit set of the group action. Vladimir used a stereographic projection to map these subsets of the 3-sphere back in Euclidean 3-dimensional space. When the set gets deformed, either the group or the stereographic projection is modified.

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