Elliptic Values of Word Maps on SL(2,R) and a Question of Calegari and Walker on Rotation Numbers

Abstract

In an Oberwolfach report of 2012, D. Calegari, reporting on joint work with A. Walker, asked whether there is a nontrivial word w in the free group F = ⟨a,b⟩ such that the lifted rotation number rot~(ρ(w)) vanishes for every homomorphism ρ from F to the universal central extension Homeo⁺(S¹)~ of the group of orientation-preserving homeomorphisms of the circle. We show that there is no such word. The main step is elementary: for every nontrivial w in the commutator subgroup of F there are A, B ∈ SL(2,R) with |tr w(A,B)| < 2. The proof combines the classical trace polynomials of Fricke and Vogt with a comparison of signs on the two sides of the surface of reducible characters. Lifting such a pair to the universal cover of PSL(2,R) gives w a rotation number that is not an integer; words outside the commutator subgroup are treated with rigid rotations. All these representations are real-analytic. For w = [a,b] the negative answer is already contained in an example of Calegari and Walker (2011). For the word w0 of length 36 printed in the report we give rational matrices A and B, both hyperbolic, with tr w0(A,B) = 0; with the lifts that have fixed points, rot~(ρ(w0)) = −1/2, and the extremal value R(w0,0,0) equals 1. We also recall, after Calegari and Walker, why rot~(ρ(w0)) = 0 whenever ρ(a) or ρ(b) is elliptic. The second question of the report, on real-analytic representations that realise the extremal rotation number R(w,r,s), is not treated. No printed answer to the first question was found in the sources we searched; this is not a claim of priority. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-12007-010 (D. Calegari, joint work with A. Walker, Ziggurats and rotation numbers, Oberwolfach Report 35/2012, pp. 2173–2175: two open questions on rotation numbers of words).