Modules Satisfying the S-Noetherian Property and S-ACCR

In this paper we show that the space of irreducible representations from a finitely presented group into the group of isometries of a rank one symmetric space of non-compact type, embeds into ℝn for some n, where the coordinates are the translation lengths of isometries in the representation. The ingredients of the proof consist of the two facts that the representation is determined by its marked length spectrum and that the nested sequence of algebraic subvarieties is stabilised at a finite step by the Noetherian property of the polynomial ring. As a minor application, we use this fact to simplify McMullen's proof about the exponential algebraic convergence of Thurston's double limit to the geometrically infinite manifold in the space of discrete faithful representations of π1(S) in Iso+.

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RESIDUAL FINITENESS AND THE NOETHERIAN PROPERTY OF FINITELY GENERATED LIE ALGEBRAS

Mathematics of the USSR-Sbornik ◽

10.1070/sm1989v064n02abeh003322 ◽

1989 ◽

Vol 64 (2) ◽

pp. 495-504 ◽

Cited By ~ 1

Author(s):

M V Zaĭtsev

Keyword(s):

Lie Algebras ◽

Residual Finiteness ◽

Finitely Generated ◽

Noetherian Property

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The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091514000303 ◽

2015 ◽

Vol 58 (3) ◽

pp. 543-580

Author(s):

V. V. Bavula

Keyword(s):

Weyl Algebra ◽

Differential Operators ◽

Polynomial Algebra ◽

Group Of Automorphisms ◽

Algebra Ii ◽

Noetherian Property

AbstractThe algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:

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Ideas from Zariski Topology in the Study of Cubical Homology

Canadian Journal of Mathematics ◽

10.4153/cjm-2007-043-3 ◽

2007 ◽

Vol 59 (5) ◽

pp. 1008-1028

Author(s):

Tomasz Kaczynski ◽

Marian Mrozek ◽

Anik Trahan

Keyword(s):

Algebraic Geometry ◽

Dynamical Systems ◽

Digital Imaging ◽

Zariski Topology ◽

Euclidean Topology ◽

Separation Axioms ◽

Cubical Sets ◽

Noetherian Property ◽

Cubical Homology

AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

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