Volume 13, Issue 2


1. Finitely generated subgroups of free groups as formal languages and their cogrowth

Arman Darbinyan ; Rostislav Grigorchuk ; Asif Shaikh.
For finitely generated subgroups H of a free group Fm of finite rank m, we study the language LH of reduced words that represent H which is a regular language. Using the (extended) core of Schreier graph of H, we construct the minimal deterministic finite automaton that recognizes LH. Then we characterize the f.g. subgroups H for which LH is irreducible and for such groups explicitly construct ergodic automaton that recognizes LH. This construction gives us an efficient way to compute the cogrowth series LH(z) of H and entropy of LH. Several examples illustrate the method and a comparison is made with the method of calculation of LH(z) based on the use of Nielsen system of generators of H.

2. A fibering theorem for 3-manifolds

Jordan A. Sahattchieve.
An erratum to this article is posted at https://gcc.episciences.org/page/errata This paper generalizes results of M. Moon on the fibering of certain compact 3-manifolds over the circle. It also generalizes a theorem of H. B. Griffiths on the fibering of certain 2-manifolds over the circle.

3. The Axiomatics of Free Group Rings

Benjamin Fine ; Anthony Gaglione ; Martin Kreuzer ; Gerhard Rosenberger ; Dennis Spellman.
In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R was examined. Here we assume that R is a commutative ring with identity 10. Of course, these are relative to an appropriate logical language L0,L1,L2 for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if R[G] is elementarily equivalent to S[H] with respect to L2, then simultaneously the group G is elementarily equivalent to the group H with respect to L0, and the ring R is elementarily equivalent to the ring S with respect to L1. We then let F be a rank 2 free group and Z be the ring of integers. Examining the universal theory of the free group ring Z[F] the hazy conjecture was made that the universal sentences true in Z[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in Z[F] modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for Z[F].