TL;DR
This paper provides explicit formulas for the fringe lengths of Ziggurat graphs related to extremal rotation numbers, revealing self-similarity patterns that explain previously observed phenomena.
Contribution
It introduces explicit formulas for Ziggurat fringe lengths and uncovers their self-similar structure, advancing understanding of bounded cohomology in free groups.
Findings
Explicit formulas for fringe lengths
Revealed partial integral projective self-similarity
Explained observed experimental phenomena
Abstract
We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko and Calegari-Walker.
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