366 research outputs found
On Quasiperiodic Morphisms
Weakly and strongly quasiperiodic morphisms are tools introduced to study
quasiperiodic words. Formally they map respectively at least one or any
non-quasiperiodic word to a quasiperiodic word. Considering them both on finite
and infinite words, we get four families of morphisms between which we study
relations. We provide algorithms to decide whether a morphism is strongly
quasiperiodic on finite words or on infinite words.Comment: 12 page
On Christoffel and standard words and their derivatives
We introduce and study natural derivatives for Christoffel and finite
standard words, as well as for characteristic Sturmian words. These
derivatives, which are realized as inverse images under suitable morphisms,
preserve the aforementioned classes of words. In the case of Christoffel words,
the morphisms involved map to (resp.,~) and to
(resp.,~) for a suitable . As long as derivatives are
longer than one letter, higher-order derivatives are naturally obtained. We
define the depth of a Christoffel or standard word as the smallest order for
which the derivative is a single letter. We give several combinatorial and
arithmetic descriptions of the depth, and (tight) lower and upper bounds for
it.Comment: 28 pages. Final version, to appear in TC
On sets of indefinitely desubstitutable words
The stable set associated to a given set S of nonerasing endomorphisms or
substitutions is the set of all right infinite words that can be indefinitely
desubstituted over S. This notion generalizes the notion of sets of fixed
points of morphisms. It is linked to S-adicity and to property preserving
morphisms. Two main questions are considered. Which known sets of infinite
words are stable sets? Which ones are stable sets of a finite set of
substitutions? While bringing answers to the previous questions, some new
characterizations of several well-known sets of words such as the set of binary
balanced words or the set of episturmian words are presented. A
characterization of the set of nonerasing endomorphisms that preserve
episturmian words is also provided
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
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
Deformation Theory and Partition Lie Algebras
A theorem of Lurie and Pridham establishes a correspondence between formal
moduli problems and differential graded Lie algebras in characteristic zero,
thereby formalising a well-known principle in deformation theory. We introduce
a variant of differential graded Lie algebras, called partition Lie algebras,
in arbitrary characteristic. We then explicitly compute the homotopy groups of
free algebras, which parametrise operations. Finally, we prove generalisations
of the Lurie-Pridham correspondence classifying formal moduli problems via
partition Lie algebras over an arbitrary field, as well as over a complete
local base.Comment: 89 page
- …