On the motivic Higman conjecture

Abstract

The G-representation variety RG(Σg) parametrizes the representations of the fundamental group π1(Σg) of a closed surface into an algebraic group G. For G the groups of n×n upper triangular matrices or unipotent upper triangular matrices, we compare two methods for computing algebraic invariants of RG(Σg). 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 RG(Σg) in the Grothendieck ring of varieties for n=1,…,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 RG(Σg) for n=1,…,10. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss how the representation varieties for the groups of unipotent upper triangular matrices are related to Higman's conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture.