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

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 \(F=\langle a,b\rangle\) such that the lifted rotation number \(\widetilde{\mathrm{rot}}(\rho(w))\) vanishes for every homomorphism \(\rho\) from \(F\) to the universal central extension \(\mathrm{Homeo}^+(S^1)^{\sim}\) 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 \(A,B\in\mathrm{SL}(2,\mathbb{R})\) with \(|\operatorname{tr} w(A,B)|<2\). 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 \(\mathrm{PSL}(2,\mathbb{R})\) 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 \(w_0\) of length \(36\) printed in the report we give rational matrices \(A\) and \(B\), both hyperbolic, with \(\operatorname{tr} w_0(A,B)=0\); with the lifts that have fixed points, \(\widetilde{\mathrm{rot}}(\rho(w_0))=-\frac12\), and the extremal value \(R(w_0,0,0)\) equals \(1\). We also recall, after Calegari and Walker, why \(\widetilde{\mathrm{rot}}(\rho(w_0))=0\) whenever \(\rho(a)\) or \(\rho(b)\) 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.