226 research outputs found
Additive decomposability of functions over abelian groups
Abelian groups are classified by the existence of certain additive
decompositions of group-valued functions of several variables with arity gap 2.Comment: 17 page
Irrationality of generic quotient varieties via Bogomolov multipliers
The Bogomolov multiplier of a group is the unramified Brauer group associated
to the quotient variety of a faithful representation of the group. This object
is an obstruction for the quotient variety to be stably rational. The purpose
of this paper is to study these multipliers associated to nilpotent pro-
groups by transporting them to their associated Lie algebras. Special focus is
set on the case of -adic Lie groups of nilpotency class , where we
analyse the moduli space. This is then applied to give information on
asymptotic behaviour of multipliers of finite images of such groups of exponent
. We show that with fixed and increasing , a positive proportion of
these groups of order have trivial multipliers. On the other hand, we
show that by fixing and increasing , log-generic groups of order
have non-trivial multipliers. Whence quotient varieties of faithful
representations of log-generic -groups are not stably rational. Applications
in non-commutative Iwasawa theory are developed.Comment: 34 pages; improved expositio
On cardinal invariants and generators for von Neumann algebras
We demonstrate how virtually all common cardinal invariants associated to a
von Neumann algebra M can be computed from the decomposability number, dec(M),
and the minimal cardinality of a generating set, gen(M). Applications include
the equivalence of the well-known generator problem, "Is every separably-acting
von Neumann algebra singly-generated?", with the formally stronger questions,
"Is every countably-generated von Neumann algebra singly-generated?" and "Is
the gen invariant monotone?" Modulo the generator problem, we determine the
range of the invariant (gen(M), dec(M)), which is mostly governed by the
inequality dec(M) leq c^{gen(M)}.Comment: 22 pages; the main additions are Theorem 3.8 and Section
List decoding group homomorphisms between supersolvable groups
We show that the set of homomorphisms between two supersolvable groups can be
locally list decoded up to the minimum distance of the code, extending the
results of Dinur et al who studied the case where the groups are abelian.
Moreover, when specialized to the abelian case, our proof is more streamlined
and gives a better constant in the exponent of the list size. The constant is
improved from about 3.5 million to 105.Comment: 11 page
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