Algebro-Geometric Invariants of Finitely Generated Groups (The Profile of a Representation Variety)

S. Liriano S. Majewicz

Published 2008-04-03Version 1

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be P_d(R_A(G))=(N_d(R_A(G)),...,N_0(R_A(G))), where N_i(R_A(G)) stands for the number of irreducible components of R_A(G) of dimension i, where 0\leq i\leq d, and d=Dim(R_A(G)). We then use this invariant in the study of fg groups and prove various results. In particular, we show that if G an orientable surface group of genus g\geq 1, then P_d(R_{SL(2,C)}(G))\neq P_d(R_{PSL(2,C)}(G)). We also show that the same holds for G a torus knot group with presentation <x,y;x^p=y^t> where both p,t are greater than 2, and that the same also holds when G is a the fundamental group of a compact non-orientable surface of genus g\geq 3. Further, we show that if a group G can be n+1 generated, and presented by <x_1,...,x_n,y ; W=y^p>, where W is a non-trivial word in F_n=<x_1,...,x_n>, and A=PSL(2, C), that then Dim(R_{A}(G)) is equal to Max{3n, Dim(R_{A}(G'))+2 \} \leq 3n+1, where G'=<x_1,...,x_n; W=1>. We also give a condition guaranteeing that the resulting algebraic variety is reducible.