73,450 research outputs found
Additive relative invariants and the components of a linear free divisor
A 'prehomogeneous vector space' is a rational representation
of a connected complex linear algebraic group
that has a Zariski open orbit . Mikio Sato showed that the
hypersurface components of are related to the rational
characters of , an algebraic abelian quotient
of . Mimicking this work, we investigate the 'additive functions' of ,
the homomorphisms . Each such is related to an
'additive relative invariant', a rational function on such that on for all . Such an is homogeneous of
degree , and helps describe the behavior of certain subsets of under the
--action.
For those prehomogeneous vector spaces with a type of hypersurface called
a linear free divisor, we prove there are no nontrivial additive functions of
, and hence is an algebraic torus. From this we gain insight into the
structure of such representations and prove that the number of irreducible
components of equals the dimension of the abelianization of . For some
special cases ( abelian, reductive, or solvable, or irreducible) we
simplify proofs of existing results. We also examine the homotopy groups of
.Comment: 27 pages. From v1, strengthen results in section 3, improve prose,
and update contact informatio
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
Foliations for solving equations in groups: free, virtually free, and hyperbolic groups
We give an algorithm for solving equations and inequations with rational
constraints in virtually free groups. Our algorithm is based on Rips
classification of measured band complexes. Using canonical representatives, we
deduce an algorithm for solving equations and inequations in hyperbolic groups
(maybe with torsion). Additionnally, we can deal with quasi-isometrically
embeddable rational constraints.Comment: 70 pages, 7 figures, revised version. To appear in Journal of
Topolog
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