arXiv subjects

S. Liriano

Publications and source records attributed to S. Liriano.

3 recordsLinked to original sources

arXiv2006

Krull dimension and deviation in certain parafree groups

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Gilbert Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the lower central sequence with a free group of rank n. The deviation of a finitely generated (fg) parafree group G is the difference between the minimum possible number of generators of G and the rank of G. Let G be a fg group, then Hom(G,SL(2, C)) inherits the structure of an algebraic variety, denoted by R(G), and known as its "representation variety". If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G))=3n. It is known that for n \ge 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n \ge 2, and k \ge 1, there exist infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, it is shown that there exist infinitely many parafree groups G of rank n with Dim(R(G))> q, where q \ge 3n is an arbitrary integer.

math.GR↗

arXiv2006

Irreducible components in an algebraic variety of representations of a family of one-relator groups

Given a finitely generated group G, the set Hom(G,SL_2 C) inherits the structure of an algebraic variety R(G)called the "representation variety" of G. This algebraic variety is an invariant of G. Let G_{pt}=< a, b; a^p= b^t>, where p, t are integers greater than one. In this paper a formula is produced yielding the number of four dimensional irreducible components of the affine algebraic variety R(G_{pt}). A direct consequence of the main theorem of this paper is that if K is a torus knot, then its genus equals the number of four dimensional components of R(G_{pt}) corresponding to its knot group G_{pt}.

math.GR↗

arXiv2006

Algebraic Geometric Invariants of Parafree Groups

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL(2,C) inherits the structure of an algebraic variety known as the "representation variety" of G. 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 one-relator parafree groups are computed. It is then shown that in an infinite number of cases these invariants successfully discriminate between isomorphism types within the class of parafree groups of the same rank. This is quite surprising, since in this paper it also shown that an n generated group G is free of rank n iff Dim(R(G))=3n. In fact, a direct consequence of Theorem 1.6 in this paper is that given an arbitrary positive integer k, and any integer r > 1 there exist infinitely many non-isomorphic (fg) parafree groups of rank r and deviation one with representation varieties of dimension 3r, having more than k mirc of dimension 3r.

math.GR↗