Krull Dimension and Deviation in Certain Parafree Groups

Abstract

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. 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 same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) − n, where μ(G) is the minimum possible number of generators of G.

Let G be fg; then Hom(G, SL ₂ℂ) inherits the structure of an algebraic variety, denoted by R(G). 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 ≥ 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 ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer.

Key Words:

• Free groups
• Invariants
• Parafree groups
• Representational Varieties

2000 Mathematics Subject Classification:

• Primary 20F22
• 20E05
• 20C99
• Secondary 14L10
• 14Q15

In memory of Marcelo Llarull (1958–2005).

Notes

The Zariski topology on ℂ ⁿ is the topology each of whose closed sets arises as the common zero locus of a set of n variate polynomials over ℂ. A subset V in ℂ ⁿ is termed an algebraic variety provided it is closed in the subspace topology induced from ℂ ⁿ on V.

A regular map is continuous morphism in the Zariski topology.

The Hilbert Basis Theorem guarantees that only a finite number of the relators of G on the generating set X are necessary in defining the algebraic variety R(G).

A free group of rank n is a parafree group of deviation 0 and rank n.

The lower central sequence of a group G is the sequence given by G/γ _n G, where γ _n G is the n-term of the lower central series.

A group is termed cyclically pinched if it can be given a presentation of the type G = ⟨ X ∪ Y; W = V ⟩, where X is a finite set of generators, W is a non-trivial word in the free group on X, and Y is a finite set of generators, and the word V a non-trivial word in the free group on Y.

In particular, k can be taken equal to N ₇(R(G)) for any fixed fg parafree group G with Dim(R(G)) = 7.

Note that if k = 1 then S contains at least one 6-dimensional component that does not coincide with the 6-dimensional component(s) in (R(G _{k, i} ) − S).

Overline stands for the Zariski closure.