64 research outputs found
Ranks of subgroups in boundedly generated groups
We show that an infinite residually finite boundedly generated group has an
infinite chain of finite index subgroups with ranks uniformly bounded, and give
(sublinear) upper bounds on the ranks of arbitrary finite index subgroups of
boundedly generated groups (examples which come close to achieving these bounds
are presented). This proves a strong form of a conjecture of Abert,
Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite
boundedly generated residually finite groups is . Furthermore, our first
result establishes a variant of a conjecture of Lubotzky on the ranks of finite
index subgroups of special linear groups over the integers, and is analogous to
a result of Pyber and Segal for solvable groups
Ascending chains of finitely generated subgroups
We show that a nonempty family of -generated subgroups of a pro- group
has a maximal element. This suggests that 'Noetherian Induction' can be used to
discover new features of finitely generated subgroups of pro- groups. To
demonstrate this, we show that in various pro- groups (e.g. free
pro- groups, nonsolvable Demushkin groups) the commensurator of a finitely
generated subgroup is the greatest subgroup of containing
as an open subgroup. We also show that an ascending sequence of
-generated subgroups of a limit group must terminate (this extends the
analogous result for free groups proved by Takahasi, Higman, and
Kapovich-Myasnikov)
Balanced presentations for fundamental groups of curves over finite fields
We show that the algebraic fundamental group of a smooth projective curve
over a finite field admits a finite topological presentation where the number
of relations does not exceed the number of generators
Virtual retraction and Howson's theorem in pro- groups
We show that for every finitely generated closed subgroup of a
non-solvable Demushkin group , there exists an open subgroup of
containing , and a continuous homomorphism satisfying
for every . We prove that the intersection of a pair of
finitely generated closed subgroups of a Demushkin group is finitely generated
(giving an explicit bound on the number of generators). Furthermore, we show
that these properties of Demushkin groups are preserved under free pro-
products, and deduce that Howson's theorem holds for the Sylow subgroups of the
absolute Galois group of a number field. Finally, we confirm two conjectures of
Ribes, thus classifying the finitely generated pro- M. Hall groups
Squarefree polynomials with prescribed coefficients
For nonempty subsets of a (large enough) finite field
satisfying we show that there exist such that is a squarefree polynomial
The Hanna Neumann conjecture for Demushkin Groups
We confirm the Hanna Neumann conjecture for topologically finitely generated
closed subgroups and of a nonsolvable Demushkin group . Namely, we
show that \begin{equation*} \sum_{g \in U \backslash G/W} \bar d(U \cap
gWg^{-1}) \leq \bar d(U) \bar d(W) \end{equation*} where and is the least cardinality of a topological
generating set for the group
On the Chowla and twin primes conjectures over
Using geometric methods, we improve on the function field version of the
Burgess bound, and show that, when restricted to certain special subspaces, the
M\"{o}bius function over can be mimicked by Dirichlet
characters. Combining these, we obtain a level of distribution close to for
the M\"{o}bius function in arithmetic progressions, and resolve Chowla's
-point correlation conjecture with large uniformity in the shifts. Using a
function field variant of a result by Fouvry-Michel on exponential sums
involving the M\"{o}bius function, we obtain a level of distribution beyond
for irreducible polynomials, and establish the twin prime conjecture in a
quantitative form. All these results hold for finite fields satisfying a simple
condition
Semi-free subgroups of a profinite surface group
We show that every closed normal subgroup of infinite index in a profinite
surface group is contained in a semi-free profinite normal subgroup of
. This answers a question of Bary-Soroker, Stevenson, and Zalesskii
M\"{o}bius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields
We establish cancellation in short sums of certain special trace functions
over below the P\'{o}lya-Vinogradov range, with savings
approaching square-root cancellation as grows. This is used to resolve the
-analog of Chowla's conjecture on cancellation in M\"{o}bius
sums over polynomial sequences, and of the Bateman-Horn conjecture in degree
, for some values of . A final application is to sums of trace functions
over primes in
Finite presentation of the tame fundamental group
Let be a prime number, and let be an algebraically closed field of
characteristic . We show that the tame fundamental group of a smooth affine
curve over is a projective profinite group. We prove that the fundamental
group of a smooth projective variety over is finitely presented. More
generally we prove that the tame fundamental group of a smooth quasi-projective
variety over which admits a good compactification is finitely presented.
v2: references added. Thank you to all for the friendly and fruitful
comments.Comment: 19 page
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