Non-injectivity of the trace map for character varieties

TL;DR

This paper proves that the Goldman trace map for surface character varieties is never injective, providing explicit kernel elements using algebraic identities and free group words, confirming a long-standing prediction.

Contribution

It demonstrates the non-injectivity of the Goldman trace map for all ranks by explicitly constructing kernel elements based on algebraic identities and free group words.

Findings

The trace map is never injective for any rank.

Explicit kernel elements are constructed using the Amitsur-Levitzki identity.

The construction confirms Goldman's predicted non-injectivity in arbitrary rank.

Abstract

Given a closed oriented surface $\Sigma$ 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 $\pi_1(\Sigma)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $\pi_1(\Sigma)$, 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.