Polynomial encoding of hyperbolic geodesic lengths ($\mathrm{tr}(\rho (w){)}^2=4{\cosh }^2({\ell }_{\rho }(w)/2)$)

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.