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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.18649 |
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| _version_ | 1866916023311007744 |
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| author | Das, Deblina Kabiraj, Arpan |
| author_facet | Das, Deblina Kabiraj, Arpan |
| contents | Given a closed oriented surface $Σ$ 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 $π_1(Σ)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $π_1(Σ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18649 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-injectivity of the trace map for character varieties Das, Deblina Kabiraj, Arpan Geometric Topology Given a closed oriented surface $Σ$ 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 $π_1(Σ)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $π_1(Σ)$, 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. |
| title | Non-injectivity of the trace map for character varieties |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2605.18649 |