Algebraic geometric invariants of parafree groups

被引:5
|
作者
Liriano, Sal [1 ]
机构
[1] CUNY City Coll, Ctr Algorithms & Interact Sci Software, New York, NY 10031 USA
关键词
parafree; representation variety; free; residually nilpotent; invariants;
D O I
10.1142/S0218196707003494
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL2C inherits the structure of an algebraic variety known as the representation variety of G in SL2C. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim(R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r >= 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.
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页码:155 / 169
页数:15
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