Algebro-Geometric Invariants of Finitely Generated Groups (The Profile of a Representation Variety)
Abstract
If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by RA(G). Here we define the profile function, Pd(RA(G)), of the representation variety of G over A to be Pd(RA(G))=(Nd(RA(G)),...,N0(RA(G))), where Ni(RA(G)) stands for the number of irreducible components of RA(G) of dimension i, where 0≤ i≤ d, and d=Dim(RA(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≥ 1, then Pd(RSL(2,C)(G))≠ Pd(RPSL(2,C)(G)). We also show that the same holds for G a torus knot group with presentation <x,y;xp=yt> 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≥ 3. Further, we show that if a group G can be n+1 generated, and presented by <x1,...,xn,y ; W=yp>, where W is a non-trivial word in Fn=<x1,...,xn>, and A=PSL(2, C), that then Dim(RA(G)) is equal to Max3n, Dim(RA(G'))+2 \ ≤ 3n+1, where G'=<x1,...,xn; W=1>. We also give a condition guaranteeing that the resulting algebraic variety is reducible.