Dimension bounds for relative character varieties on the projective line with three punctures $G=GL(r),O(r),Sp(r)$

We consider relative character varieties on ${\mathbb{P}}^1\mathrm{\backslash }\{0,1,\mathrm{\infty }\}$ with $G=GL(r),O(r)$, or $Sp(r)$. Using a diagrammatic method of Simpson’s, we give an explicit linear upper bound $R(d)$ on the rank $r$ of an MC-minimal character variety of dimension $d>2$. An arbitrary character variety is isomorphic, via Katz’s middle convolution, to one satisfying the bound. For the general linear and non-overlapping quadratic cases, the bounds we give are the sharpest possible using this method.