Arnaldo Mandel

GRAPH-ENCODED 3-MANIFOLDS

Abstract From each G in a certain class of 4-regular edge-colored graphs we obtain a ball complex whose underlying space | G | is a 3-manifold and which has G as its 1-skeleton. The construction of this ball complex B ( G ) is dual to the one of Crystallization Theory originated by [Pezzana, 1974] and [Ferri, 1976]. There a pseudo-complex K ( G ) is obtained from an edge-colored graph G . If | K ( G )| is a manifold, then K ( G ) and B ( G ) are dual cellular decomposition of the same manifold, | G |≅| K ( G )|. The proof that every 3-manifold can be represented by a | G | is somewhat simpler than the original based on K ( G ): it is a direct consequence of the triangulation theorem [Moise, 1977]. We present new crystallizations for some of the most commonly studied 3-manifolds. They have fewer vertices and more symmetry than previously published ones. A computer program was used to obtain them from bigger 3-gems and another to get symmetric drawings by using the algorithm in [Tutte, 1963]. A 4-parametric family of 3-manifolds which includes the lens spaces and Poincares sphere is introduced. A table of the homologies of an initial segment of this family which exhibits its richness appears in the Appendix.

On finite semigroups of matrices

Abstract Finite semigroups of n by n matrices over the naturals are characterized both by algebraic and combinatorial methods. Next we show that the cardinality of a finite semigroup S of n by n matrices over a field is bounded by a function depending only on n , the number of generators of S and the maximum cardinality of its subgroups. As a consequence, given n and k , there exist, up to isomorphism, only a finite number of finite semigroups of n by n matrices over the rationals, generated by at most k elements. Among other applications to Automaton Theory, we show that it is decidable whether the behavior of a given N – Σ automaton is bounded.

Are there free groups in division rings

The following conjecture is investigated: a noncentral subnormal subgroup of the multiplicative group of a division ring contains a noncyclic free subgroup. Special cases are proved, entailing several known commutativity theorems. Also a new framework is presented for some kinds of commutativity theorems, based on the existence of (group) words for which one can always find an appropriate substitution by elements of such a subnormal subgroup that yields a noncentral element. Several families of such words are given; one gets commutativity theorems imposing some restrictions (like periodicity) to the image of these words.

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

FREE SUBGROUPS IN THE GROUP OF UNITS OF A TWISTED GROUP ALGEBRA

Consider a twisted group algebra K t G of a torsion group G over a field K. Provided K is sufficiently large, then either K t G is commutative or U(K t G) contains a noncommutative free group. *Research partially supported by CNPq. †Research partially supported by CNPq, by FAPESP Proc. 96/4505–2 and MCT/FINEP, Projeto PRONEX 107/97.

From monomials to words to graphs

Given a finite alphabet X and an ordering ≺ on the letters, the map σ≺ sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Grobner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal 〈σ≺(I)〉 generated by σ≺(I) in the free monoid is finitely generated. Whether there exists an ≺ such that 〈σ≺(I)〉 is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.

A commutativity theorem for division rings and an extension of a result of Faith

It is shown that if a division ring is such that for every element a there exist co-monic integer polynomials fa, Ga of different orders so that fa(a)/Ga(a) is central then the ring is commutative. We extend this result to reduced PI rings, and show that it breaks down slightly for primitive rings, where the exceptions are characterized. This extends some earlier theorems of Herstein and Faith.