Non-injectivity of the trace map for character varieties

Given a closed oriented surface $\mathrm{\Sigma }$ of genus at least two, the Goldman trace map defines a function from the vector space generated by the free homotopy classes of oriented closed curves to the Poisson algebra of regular functions on the $G$-character variety where $G$ is a reductive (real or complex) linear Lie group. In this article, we prove that this map is never injective. For each $n$, we construct an explicit nonzero element of the vector space whose associated trace function vanishes on every homomorphism from ${\pi }_1(\mathrm{\Sigma })$ to $G{L}_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of ${\pi }_1(\mathrm{\Sigma })$, ensuring that no cancellation occurs at the level of free homotopy classes. This gives a uniform family of explicit kernel elements, proving Goldman’s predicted non-injectivity of the trace map in arbitrary rank.