An Explicit Equation for the $\mathrm{SL}_2(\mathbb{C})$ Character Variety of $\mathbb{RP}^2 \# \mathbb{RP}^2 \# \mathbb{RP}^2$

Abstract

Let $N$ denote the closed non-orientable surface of Euler characteristic $-1$, i.e. the connected sum of three real projective planes $N = \mathbb{RP}^2 \# \mathbb{RP}^2 \# \mathbb{RP}^2$, with fundamental group presented as $\langle A,B,C \mid A^2 B^2 C^2 = I \rangle$ 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 $\mathrm{SL}_2(\mathbb{C})$ GIT character variety $X(N)$. We prove that the trace coordinates $a = \mathrm{tr}(A)$, $b = \mathrm{tr}(B)$, $c = \mathrm{tr}(C)$, $p = \mathrm{tr}(AB)$ 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 $c \neq 0$ it is isomorphic to the image of $X(N)$ under the trace map, with the remaining trace coordinates $\mathrm{tr}(BC)$, $\mathrm{tr}(CA)$, $\mathrm{tr}(ABC)$ 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.