66 research outputs found
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
When does the associated graded Lie algebra of an arrangement group decompose?
Let \A be a complex hyperplane arrangement, with fundamental group G and
holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum
possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on
the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes
(in degrees 2 and higher) as a direct product of free Lie algebras. In
particular, the ranks of the lower central series quotients of the group are
given by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We
illustrate this new Lower Central Series formula with several families of
examples.Comment: 14 pages, accepted for publication by Commentarii Mathematici
Helvetic
On derived-indecomposable solutions of the Yang--Baxter equation
If is a finite non-degenerate set-theoretic solution of the
Yang--Baxter equation, the additive group of the structure skew brace
is an -group, i.e. a group whose elements have finitely many conjugates.
Moreover, its multiplicative group is virtually abelian, so it is also close to
an -group itself. If one additionally assumes that the derived solution of
is indecomposable, then for every element of there are
finitely many elements of the form and , with . This
naturally leads to the study of a brace-theoretic analogue of the class of
-groups. For this class of skew braces, the fundamental results and their
connections with the solutions of the YBE are described: we prove that they
have good torsion and radical theories and they behave well with respect to
certain nilpotency concepts and finite generation.Comment: 24 pages. Accepted for publication in Publicacions Matem\`atique
Integral motives, relative Krull-Schmidt principle, and Maranda-type theorems
In the present article we investigate properties of the category of the
integral Grothendieck-Chow motives over a field. We discuss the Krull-Schmidt
principle for integral motives, provide a complete list of the generalized
Severi-Brauer varieties with indecomposable integral motive, and exploit a
relation between the category of motives of twisted flag varieties and integral
-adic representations of finite groups
Formality and finiteness in rational homotopy theory
We explore various formality and finiteness properties in the differential
graded algebra models for the Sullivan algebra of piecewise polynomial rational
forms on a space. The 1-formality property of the space may be reinterpreted in
terms of the filtered and graded formality properties of the Malcev Lie algebra
of its fundamental group, while some of the finiteness properties of the space
are mirrored in the finiteness properties of algebraic models associated with
it. In turn, the formality and finiteness properties of algebraic models have
strong implications on the geometry of the cohomology jump loci of the space.
We illustrate the theory with examples drawn from complex algebraic geometry,
actions of compact Lie groups, and 3-dimensional manifolds.Comment: 81 pages; prepared for a volume dedicated to Dennis Sullivan's 80th
birthda
A notion of geometric complexity and its application to topological rigidity
We introduce a geometric invariant, called finite decomposition complexity
(FDC), to study topological rigidity of manifolds. We prove for instance that
if the fundamental group of a compact aspherical manifold M has FDC, and if N
is homotopy equivalent to M, then M x R^n is homeomorphic to N x R^n, for n
large enough. This statement is known as the stable Borel conjecture. On the
other hand, we show that the class of FDC groups includes all countable
subgroups of GL(n,K), for any field K, all elementary amenable groups, and is
closed under taking subgroups, extensions, free amalgamated products, HNN
extensions, and direct unions.Comment: 58 pages, 5 figure
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