An Explicit Equation for the SL2(ℂ) Character Variety of ℝℙ2 # ℝℙ2 # ℝℙ2
Abstract
Let N denote the closed non-orientable surface of Euler characteristic −1, i.e. the connected sum of three real projective planes N = ℝℙ2 # ℝℙ2 # ℝℙ2, with fundamental group presented as ⟨ A,B,C | A2 B2 C2 = I ⟩ via the standard symmetric cross-cap model. Building on the classical trace-coordinate methods of Fricke, Vogt, and Goldman for free groups, and on the qualitative study of representation varieties of non-orientable surfaces by Benyash-Krivets and Chernousov, we give an explicit affine equation for a dense open piece of the SL2(ℂ) GIT character variety X(N). We prove that the trace coordinates a = tr(A), b = tr(B), c = tr(C), p = tr(AB) satisfy the polynomial relation c2 = abp − a2 − b2 + 4, that this hypersurface is smooth and irreducible, and that on the open locus c ≠ 0 it is isomorphic to the image of X(N) under the trace map, with the remaining trace coordinates tr(BC), tr(CA), tr(ABC) recovered as regular functions on this locus. We further identify the boundary locus c = 0 with representations satisfying the matrix identity A2 B2 = −I, and describe its representation-theoretic structure, while leaving a complete global classification along this locus as an open problem. All key trace identities are proved from Cayley-Hamilton first principles and independently checked numerically against explicit matrices.
Related
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- Character sums on an oriented singer conic and explicit Ramanujan double covers
BibTeX
@misc{chen2026explicit,
title = {{An Explicit Equation for the $\mathrmSL_2(\mathbbC)$ Character Variety of $\mathbbRP^2 \# \mathbbRP^2 \# \mathbbRP^2$}},
author = {Yinxi Chen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08513},
eprint = {2611.08513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08513},
}
RIS
TY - UNPB
TI - An Explicit Equation for the SL_2(ℂ) Character Variety of ℝℙ^2 # ℝℙ^2 # ℝℙ^2
AU - Yinxi Chen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.08513
UR - https://arxiv.org/abs/2611.08513
AB - Let N denote the closed non-orientable surface of Euler characteristic −1, i.e. the connected sum of three real projective planes N = ℝℙ^2 # ℝℙ^2 # ℝℙ^2, with fundamental group presented as ⟨ A,B,C | A^2 B^2 C^2 = I ⟩ via the standard symmetric cross-cap model. Building on the classical trace-coordinate methods of Fricke, Vogt, and Goldman for free groups, and on the qualitative study of representation varieties of non-orientable surfaces by Benyash-Krivets and Chernousov, we give an explicit affine equation for a dense open piece of the SL_2(ℂ) GIT character variety X(N). We prove that the trace coordinates a = tr(A), b = tr(B), c = tr(C), p = tr(AB) satisfy the polynomial relation c^2 = abp − a^2 − b^2 + 4, that this hypersurface is smooth and irreducible, and that on the open locus c ≠ 0 it is isomorphic to the image of X(N) under the trace map, with the remaining trace coordinates tr(BC), tr(CA), tr(ABC) recovered as regular functions on this locus. We further identify the boundary locus c = 0 with representations satisfying the matrix identity A^2 B^2 = −I, and describe its representation-theoretic structure, while leaving a complete global classification along this locus as an open problem. All key trace identities are proved from Cayley-Hamilton first principles and independently checked numerically against explicit matrices.
ER -