Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups

Abstract

We study the singular loci of the reduced $\SL_2(\CC)$-representation varieties $R_{m,n}$ of non-Hopfian Baumslag--Solitar groups $B(m,n)=\langle a,b\mid b^{-1}a^m b=a^n\rangle$. We determine their dimensions and profile functions, which list the numbers of irreducible components in descending order of dimension. The components have dimensions three or five. Their reduced singular loci have the explicit description \[ \Sing R_{m,n} =\left\{(\varepsilon I,Y): \varepsilon^m=\varepsilon^n,\quad (\tr Y)^2=\frac{(m+n)^2}{mn}\right\}, \] where $\varepsilon\in\{\pm1\}$ and $Y\in\SL_2(\CC)$. Each is a disjoint union of $μ$ copies of the smooth affine quadric $Q=\{(x,y,z)\in\CC^3:x^2+yz=1\}$, where $μ=2$ for $m-n$ odd and $μ=4$ for $m-n$ even. Thus its complex algebraic isomorphism type depends only on this parity. Each quadric is homotopy equivalent to $S^2$, so the singular locus has integral homology $\ZZ^μ$ in degrees zero and two and zero in all other degrees. An algebraic stratification gives its compactly supported E-polynomial $μ(q^2+q)$, with $q=uv$. At every singular point, $R_{m,n}$ is locally analytically the product of a smooth surface and a nodal curve.