A free subgroup in the image of the 4-strand Burau representation
Stefan Witzel, Matthew C. B. Zaremsky
TL;DR
This paper proves that the elements f^3 and k^3 generate a free subgroup in the 4-strand Burau representation by analyzing their action as isometries of a Euclidean building, using two different proof methods.
Contribution
It establishes a new free subgroup generated by specific powers of matrices in the 4-strand Burau representation, advancing understanding of its faithfulness.
Findings
f^3 and k^3 generate a free group
Two different proof techniques confirm the result
Supports the faithfulness of the Burau representation for 4 strands
Abstract
It is known that the Burau representation of the 4-strand braid group is faithful if and only if certain matrices f and k generate a (non-abelian) free group. Regarding f and k as isometries of a euclidean building we show that f^3 and k^3 generate a free group. We give two proofs, one utilizing the metric geometry of the building, and the other using simplicial retractions.
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