Markov moves, L^2-Burau maps and Lehmer's constants
(2021) Cornell University. City and Regional Planning Publications. Occasional Papers — Vol. np, n° np, p. 26 (2021)
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- We study the effect of Markov moves on L^2-Burau maps of braids, in order to construct link invariants from these maps with a process mirroring the well-known Alexander-Burau formula. We prove such a Markov invariance for the L^2-Burau maps which descend to the groups of the braid closures or lower, and for which the associated link invariants are twisted L^2-Alexander torsions. When the L^2-Burau map descends to a link group, the corresponding link invariant was known to be the L^2-Alexander torsion of the link by a previous result of A. Conway and the author. Furthermore, we find two counter-examples to Markov invariance, meaning two families of L^2-Burau maps that cannot yield link invariants with the process described in our paper. The proofs use relations between Fuglede-Kadison determinants, Mahler measures, and random walks on Cayley graphs, as well as works of Boyd, Bartholdi and Dasbach-Lalin. Along the way, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups. As a consequence, we partially answer a question of Lück, as we provide new upper bounds for Lehmer's constants for all torsionfree groups which have non-cyclic free subgroups. Our results suggest that twisted L^2-Alexander torsions are the only link invariants we can hope to construct from L^2-Burau maps with the present approach.
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Ben Aribi, F. (2021). Markov moves, L^2-Burau maps and Lehmer’s constants. Cornell University. City and Regional Planning Publications. Occasional Papers, np(np), 26. https://hdl.handle.net/2078.5/22630 (Original work published 2021)