Motivic Higman's Conjecture

The $G$-representation variety $R_G({\mathrm{\Sigma }}_g)$ parametrizes the representations of the fundamental groups of surfaces ${\pi }_1({\mathrm{\Sigma }}_g)$ into an algebraic group $G$. Taking $G$ to be the groups of $n\times n$ upper triangular or unipotent matrices, we compare two methods for computing algebraic invariants of $R_G({\mathrm{\Sigma }}_G)$. Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of $R_G({\mathrm{\Sigma }}_g)$ in the Grothendieck ring of varieties for $n=1,\dots ,5$. Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the $E$-polynomials of $R_G({\mathrm{\Sigma }}_g)$ for $n=1,\dots ,10$. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss the relation between the representation varieties of the group of unipotent matrices and Higman’s conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture.