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

TL;DR

This paper introduces the profile function as an invariant of representation varieties of finitely generated groups, revealing distinctions between groups like surface groups and knot groups through algebraic geometric properties.

Contribution

It defines the profile function for representation varieties and applies it to differentiate algebraic invariants of various finitely generated groups, including surface and knot groups.

Findings

Profile function distinguishes between surface and knot groups.

Representation variety dimensions are explicitly computed for certain groups.

Conditions for reducibility of algebraic varieties are established.

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 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.