The Bruhat-Tits tree of SL(2), Cubic Julia sets, and Thompson groups.

This morning I watch this video of James Arthur talking about a poster about the history of the Langlands programme:

I was struck by the shape of the Bruhat-Tits building in the background of the image. It’s topologically isotropic, which is a feature of (filled) Julia sets.

Bill Casselman wrote The Bruhat-Tits tree of SL(2) in which there are pictures of it.

Remark one: there is a topological embedding of the Bruhat-Tits tree of SL(2) into a cubic Julia set:

Remark two: there is a Thompson group which corresponds to the symmetries of the Bruhat-Tits tree of SL(2),
of which here is depicted one transformation thereof:

The following papers about the automorphisms of the Bruhat-Tits tree may be relevant:
Groups of hierarchomorphisms of trees and related
Hilbert spaces
by Yurii A. Neretin, though it is not clear at all whether the hierarchomorphisms he speaks of are akin to the symmetry operation visualized above, because the one visualized above requires no gluing or cutting at all. There are a few other hits on google for “”bruhat-tits tree” and “julia set””, but only 13 documents match total. Jim Belk’s paper A Thompson Group for the Basilica may also be relevant.

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