102,145 research outputs found
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
Every group is the outer automorphism group of an HNN-extension of a fixed triangle group
Fix an equilateral triangle group
with arbitrary. Our main result is: for every presentation
of every countable group there exists an HNN-extension
of such that . We construct the HNN-extensions explicitly, and examples are given. The
class of groups constructed have nice categorical and residual properties. In
order to prove our main result we give a method for recognising malnormal
subgroups of small cancellation groups, and we introduce the concept of
"malcharacteristic" subgroups.Comment: 39 pages. Final version, to appear in Advances in Mathematic
Horizontal non-vanishing of Heegner points and toric periods
Let be a totally real field and a modular \GL_2-type
abelian variety over . Let be a CM quadratic extension. Let be
a class group character over such that the Rankin-Selberg convolution
is self-dual with root number . We show that the number of
class group characters with bounded ramification such that increases with the absolute value of the discriminant of .
We also consider a rather general rank zero situation. Let be a
cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let
be a Hecke character over such that the Rankin-Selberg convolution
is self-dual with root number . We show that the number of
Hecke characters with fixed -type and bounded ramification such
that increases with the absolute value of the
discriminant of .
The Gross-Zagier formula and the Waldspurger formula relate the question to
horizontal non-vanishing of Heegner points and toric periods, respectively. For
both situations, the strategy is geometric relying on the Zariski density of CM
points on self-products of a quaternionic Shimura variety. The recent result
\cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental
to the approach.Comment: Adv. Math., to appear. arXiv admin note: text overlap with
arXiv:1712.0214
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Integrability Criterion for Abelian Extensions of Lie Groups
We establish a criterion for when an abelian extension of
infinite-dimensional Lie algebras integrates to a corresponding Lie group
extension of by , where is a connected, simply connected
Lie group and is a quotient of its Lie algebra by some discrete subgroup.
When is non-simply connected, the kernel is replaced by a central
extension of by .Comment: 11 pages, 2 figure
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