64 research outputs found
Filtrations of free groups as intersections
For several natural filtrations of a free group S we express the n-th term of
the filtration as the intersection of all kernels of homomorphisms from S to
certain groups of upper-triangular unipotent matrices. This generalizes a
classical result of Grun for the lower central filtration. In particular, we do
this for the n-th term in the lower p-central filtration of S
The lower -central series of a free profinite group and the shuffle algebra
For a prime number and a free profinite group on the basis , let
, be the lower -central filtration of . For
, we give a combinatorial description of
in terms of the Shuffle algebra on
The Zassenhaus filtration, Massey Products, and Representations of Profinite Groups
We consider the p-Zassenhaus filtration (G_n) of a profinite group G. Suppose
that G=S/N for a free profinite group S and a normal subgroup N of S contained
in S_n. Under a cohomological assumption on the n-fold Massey products (which
holds e.g., if the p-cohomological dimension of G is at most 1), we prove that
G_{n+1} is the intersection of all kernels of upper-triangular unipotent
(n+1)-dimensional representations of G over \mathbb F_p. This extends earlier
results by Minac, Spira, and the author on the structure of absolute Galois
groups of fields.Comment: Added more references, strengthened Lemma 2.3, added Remark 12.
The Cohomology of canonical quotients of free groups and Lyndon words
For a prime number and a free profinite group , let be the
th term of its lower -central filtration, and the
corresponding quotient. Using tools from the combinatorics of words, we
construct a canonical basis of the cohomology group
, which we call the Lyndon basis, and use it to
obtain structural results on this group. We show a duality between the Lyndon
basis and canonical generators of . We prove that the
cohomology group satisfies shuffle relations, which for small values of
fully describe it.Comment: Several minor issues fixed and a few references added. To appear in
Documenta Mathematic
Small Galois groups that encode valuations
Let be a prime number and let be a field containing a root of unity
of order . We prove that a certain very small canonical Galois group
over encodes the valuations on whose value group is not
-divisible and which satisfy a variant of Hensel's lemma.Comment: Final version. To appear in Acta Arithmetic
Triple Massey products and absolute Galois groups
Let be a prime number, a field containing a root of unity of order
, and the absolute Galois group. Extending results of Hopkins,
Wickelgren, Minac and Tan, we prove that the triple Massey product
contains whenever it is nonempty. This gives a new
restriction on the possible profinite group structure of .Comment: A first version of this paper, by the second-named author, was
earlier posted as arXiv:1411.4146. The current version gives a purely
group-cohomological proof of the main result. To appear in the Journal of the
European Mathematical Societ
Galois groups and cohomological functors
Let be a prime power, a field containing a root of unity of order
, and its absolute Galois group. We determine a new canonical quotient
of which encodes the full mod- cohomology
ring and is minimal with respect to this property. We
prove some fundamental structure theorems related to these quotients. In
particular, it is shown that when is an odd prime, is the
compositum of all Galois extensions of such that is
isomorphic to , or to the nonabelian group of
order and exponent .Comment: AMS-LaTeX, 29 pages. To appear in the Transactions of the American
Mathematical Societ
On the descending central sequence of absolute Galois groups
Let be an odd prime number and a field containing a primitive th
root of unity. We prove a new restriction on the group-theoretic structure of
the absolute Galois group of . Namely, the third subgroup
in the descending -central sequence of is the intersection of all open
normal subgroups such that is 1, , or the modular
group of order .Comment: We implemented the referee's comments. The paper will appear in The
American Journal of Mathematic
Filtrations of free groups arising from the lower central series
We make a systematic study of filtrations of a free group F defined as
products of powers of the lower central series of F. Under some assumptions on
the exponents, we characterize these filtrations in terms of the group algebra,
the Magnus algebra of non-commutative power series, and linear representations
by upper-triangular unipotent matrices. These characterizations generalize
classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as
well as later results on the lower p-central filtration and the p-Zassenhaus
filtrations. We derive alternative recursive definitions of such filtrations,
extending results of Lazard. Finally, we relate these filtrations to Massey
products in group cohomology
Invitation to higher local fields, Part II, section 7: Recovering higher global and local fields from Galois groups - an algebraic approach
A main problem in Galois theory is to characterize the fields with a given
absolute Galois group. We apply a K-theoretic method for constructing
valuations to study this problem in various situations. As a first application
we obtain an algebraic proof of the 0-dimensional case of Grothendieck's
anabelian conjecture (proven by Pop), which says that finitely generated
infinite fields are determined up to purely inseparable extensions by their
absolute Galois groups. As a second application (which is a joint work with
Fesenko) we analyze the arithmetic structure of fields with the same absolute
Galois group as a higher-dimensional local field.Comment: For introduction and notation, see math.NT/0012131 . Published by
Geometry and Topology Monographs at
http://www.maths.warwick.ac.uk/gt/GTMon3/m3-II-7.abs.htm
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