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

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

2008-04-03

arxiv.org/abs/0804.0460v1

Mathematics

Group Theory

22 pages

Scientific paper

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 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 , where W is a non-trivial word in F_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'=. We also give a condition guaranteeing that the resulting algebraic variety is reducible.

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