= Polynomial encoding of hyperbolic geodesic lengths
{title2=$\operatorname{tr}(\rho(w))^2=4\cosh^2(\ell_\rho(w)/2)$}
For a determinant-one lift of a <hyperbolic holonomy representation>, the squared <matrix trace> of each fixed group word is a <polynomial> in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of <affine algebraic sets>.
Back to article page
