102 research outputs found
Automorphism groups of polycyclic-by-finite groups and arithmetic groups
We show that the outer automorphism group of a polycyclic-by-finite group is
an arithmetic group. This result follows from a detailed structural analysis of
the automorphism groups of such groups. We use an extended version of the
theory of the algebraic hull functor initiated by Mostow. We thus make
applicable refined methods from the theory of algebraic and arithmetic groups.
We also construct examples of polycyclic-by-finite groups which have an
automorphism group which does not contain an arithmetic group of finite index.
Finally we discuss applications of our results to the groups of homotopy
self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of
arithmeticity results of Sullivan in rational homotopy theory
Virtually abelian K\"ahler and projective groups
We characterise the virtually abelian groups which are fundamental groups of
compact K\"ahler manifolds and of smooth projective varieties. We show that a
virtually abelian group is K\"ahler if and only if it is projective. In
particular, this allows to describe the K\"ahler condition for such groups in
terms of integral symplectic representations
Phantom Maps and Homology Theories
We study phantom maps and homology theories in a stable homotopy category S
via a certain Abelian category A. We express the group P(X,Y) of phantom maps X
-> Y as an Ext group in A, and give conditions on X or Y which guarantee that
it vanishes. We also determine P(X,HB). We show that any composite of two
phantom maps is zero, and use this to reduce Margolis's axiomatisation
conjecture to an extension problem. We show that a certain functor S -> A is
the universal example of a homology theory with values in an AB 5 category and
compare this with some results of Freyd.Comment: 25 pages, AMSLaTeX, to appear in Topolog
The fibre of a pinch map in a model category
In the category of pointed topological spaces, let F be the homotopy fibre of the
pinching map X âȘ CA â X âȘ CA/ X from the mapping cone on a cofibration A â X
onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497â520, 1973) proved
that F is weakly homotopy equivalent to the reduced product (X, A)â. In this paper
we prove an analogue of this phenomenon in a model category, under suitable
conditions including a cube axiom.Web of Scienc
Global Dimension of Polynomial Rings in Partially Commuting Variables
For any free partially commutative monoid , we compute the global
dimension of the category of -objects in an Abelian category with exact
coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to
polynomial rings in partially commuting variables.Comment: 11 pages, 2 figure
Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology
In this paper we determine the representation type of some algebras of
infinite matrices continuously controlled at infinity by a compact metrizable
space. We explicitly classify their finitely presented modules in the finite
and tame cases. The algebra of row-column-finite (or locally finite) matrices
over an arbitrary field is one of the algebras considered in this paper, its
representation type is shown to be finite.Comment: 33 page
DG-algebras and derived A-infinity algebras
A differential graded algebra can be viewed as an A-infinity algebra. By a
theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a
minimal A-infinity algebra. We introduce the notion of a derived A-infinity
algebra and show that any dga A over an arbitrary commutative ground ring k is
equivalent to a minimal derived A-infinity algebra. Such a minimal derived
A-infinity algebra model for A is a k-projective resolution of the homology
algebra of A together with a family of maps satisfying appropriate relations.
As in the case of A-infinity algebras, it is possible to recover the dga up
to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the
structure we are describing provides a complete description of the
quasi-isomorphism type of the dga.Comment: v3: 27 pages. Minor corrections, to appear in Crelle's Journa
- âŠ