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Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Cited by 5 (1 self)
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
ALGEBRAIC GEOMETRIC INVARIANTS OF PARAFREE GROUPS
, 2006
"... 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 l ..."
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Cited by 3 (2 self)
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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 isomorphism 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 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 ≥ 2 there exist infinitely many nonisomorphic fg parafree groups of rank r and deviation one 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 onerelator groups. General Structure of the Paper. This paper begins with an introduction where relevant ideas to what will follow are developed. It then goes on to define what a parafree group is, and how the notions that inspired G. Baumslag to give rise to such groups arose in the context of investigations conducted by W. Magnus, and a question of Hanna Neumann. The new results in this paper are Theorem 1.0, Theorem 1.1, Theorem 1.2, Corollary 1.2, Theorem 1.3, Theorem 1.5, and Theorem 1.6. In Section One several results from the author’s earlier work are introduced. In Section Two the following theorems are proven: 1.0, 1.1, and 1.6. The paper ends with a list notation.
KRULL DIMENSION AND DEVIATION IN CERTAIN PARAFREE GROUPS
, 2006
"... Abstract. Hanna Neumann asked whether it was possible for two nonisomorphic residually nilpotent groups, one of them free, to share the lower central sequence. G. 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 ..."
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Abstract. Hanna Neumann asked whether it was possible for two nonisomorphic residually nilpotent groups, one of them free, to share the lower central sequence. G. 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 is the difference µ(G) − µ ( G), where µ(G) is the minimum γ2G possible number of generators of G, and γ2G is the second term of the lower central series of G. Let G be a finitely generated group (fg), then Hom(G, SL(2, C)) inherits the structure of an algebraic variety, denoted by R(G), which is an invariant of fg presentations of 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 nonisomorphic 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 nonisomorphic 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.