Let A be an R G-module, where R is an integral domain and G is a soluble group. Suppose that C G(A) = 1 and A/C A(G) is not a noetherian R-module. Let L nnd(G) be the family of all subgroups H of G such that A/C A(H) is not a noetherian R-module. In this paper we study the structure of those G for which L nnd(G) satisfies the maximal condition.
With this work and its sequel, Moduli of unipotent representations II, we initiate a study of the finite dimensional algebraic representations of a unipotent group over a field of characteristic zero from the modular point of view. Let be such a group. The stack of all representations of dimension is badly behaved. In this first installment, we introduce a nondegeneracy condition which cuts out a substack which is better behaved, and, in particular, admits a coarse algebraic space, which...
Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely...
We show that the class of groups which have monoid presentations by means of finite special -confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.
We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.
It is well known that the monoid ring of the free product of a free group and a free monoid over a skew field is a fir. We give a proof of this fact that is more direct than the proof in the literature.