| Let G be a finitely generated group. The set of homomorphisms from G into SL(2,{dollar}doubc{dollar}) inherits the structure of a complex affine variety denoted by R(G). R(G) is an invariant of the group G. In the sixties G. Baumslag created a class of groups he termed Parafree. A group G is Parafree of rank k if (1) G is residually nilpotent (the lower central series intersects in the Identity). (2) There exists a free group F of rank k with {dollar}{lcub}rm Govergammasb{lcub}n{rcub}G{rcub}cong{lcub}rm Fovergammasb{lcub}n{rcub}F{rcub}{dollar} for all n {dollar}ge{dollar} 1. Where {dollar}gammasb{lcub}n{rcub}{dollar}G denotes the {dollar}rm nsp{lcub}th{rcub}{dollar} element of the lower central series. Notice that free groups are Parafree. The following groups{dollar}{dollar}rm Gsb{lcub}pqt{rcub} = langle xsb1,xsb2,xsb3; xsbsp{lcub}1{rcub}{lcub}p{rcub}xsbsp{lcub}2{rcub}{lcub}q{rcub} = xsbsp{lcub}3{rcub}{lcub}t{rcub}rangle{dollar}{dollar}are also Parafree of rank 2, for p, q, t {dollar}>1{dollar} and with no common devisors other than one. More generally, the groups{dollar}{dollar}rm Gpsb1... psb{lcub}n{rcub} = langle asb1... asb{lcub}n{rcub}; asbsp{lcub}1{rcub}{lcub}psb1{rcub}asbsp{lcub}2{rcub}{lcub}psb2{rcub}... asbsp{lcub}n-1{rcub} {lcub}psb{lcub}n-1{rcub}{rcub} = asbsp{lcub}n{rcub}{lcub}psb{lcub}n{rcub}{rcub}rangle{dollar}{dollar}are Parafree of rank n-1, when {dollar}rm psb1,psb2, ..., psb{lcub}n-1{rcub},psb{lcub}n{rcub}{dollar} have no common devisors except one. In my thesis the next two theorems are proven.; Theorem 1. Dim R(G{dollar}sb{lcub}rm pqt{rcub}{dollar}) = 6, and R(G{dollar}sb{lcub}rm pqt{rcub}{dollar}) is a reducible variety.; Theorem 2. For n {dollar}ge{dollar} 3, Dim(Gp{dollar}sb1...rm psb{lcub}n{rcub}) = 3(n-1).{dollar}; Theorems 1 and 2 are especially striking when one learns that R(G), for G free of rank n is an irreducible affine variety of dimension 3n. Wilhelm Magnus, first in a paper titled "The Uses of Two by Two Matrices in Combinatorial Group Theory", and then the joint book with the mathematics historian Bruce Chandler "The History of Combinatorial Group Theory" (1982) asks whether any of the Parafree groups{dollar}{dollar}rm Ssbsp{lcub}i{rcub}{lcub}*{rcub},j = a langle a,b,c; a = lbrack a,csp{lcub}i{rcub}rbrack lbrack b,c,sp{lcub}j{rcub}rbrackrangle{dollar}{dollar}are ever different for different integer parameters i, j {dollar}>{dollar} 1. In the thesis the following is shown:; Theorem 3. The groups {dollar}rm Ssbsp{lcub}1,1{rcub}{lcub}*{rcub}{dollar} and {dollar}rm Ssbsp{lcub}30,30{rcub}{lcub}*{rcub}{dollar} are not isomorphic.; Other results appear in the thesis; we mention a few: (1) A proof that free groups are Hopfian using naive algebraic Geometry. (2) A proof that if G is a group having a presentation with n generators and m relations, and also a presentation with n{dollar}-{dollar}m generators, that then G is free.; The proof of 2 was effected using naive arguments from algebraic geometry. In the book "The History of Combinatorial Group Theory" we are told that Magnus proved this theorem using lower central series arguments in 1939, and that in 1967 U. Stammbach gave a proof involving homological methods. |
