Low-dimensional Affine Synchronizing Groups

100 Pages·2012·1.18 MB·English

Universidade de Lisboa
Faculdade de Ciˆencias
Departamento de Matema´tica
Low-dimensional Affine Synchronizing
Groups
Ana Filipa Costa da Silva
Disserta¸ca˜o
Mestrado em Matema´tica
2012

Universidade de Lisboa
Faculdade de Ciˆencias
Departamento de Matema´tica
Low-dimensional Affine Synchronizing
Groups
Ana Filipa Costa da Silva
Disserta¸ca˜o
Mestrado em Matema´tica
Orientador: Doutor Csaba Schneider
2012

i
Abstract
The synchronization property emerged from finite state automata and
transformation semigroup theory. Synchronizing permutation groups were
ˇ
introduced by Arnold and Steinberg to study the Cerny´ Conjecture. In this
thesis we study the synchronization property in affine permutation groups
of low-dimensions. J.E. Pin proved that one-dimensional affine groups are
synchronizing. Hence our main results concern affine groups in dimension 2.
We used the characterization given by Neumann of synchronization using
graph theory, which relies on the study of the equality between the clique
number and the chromatic number of certain graphs invariant under the
actions of a group, called the suitability property. It turned out that some of
such graphs for two-dimensional affine groups have an interesting geometry
and are part of a widely studied class of graphs, the generalized Paley graphs.
Further, we used the properties of the theta-function defined by Lov´asz
connected to eigenvalues of a graph to obtain a necessary condition for the
suitability in edge-transitive and vertex-transitive graphs. In this thesis we
stated a criterion to decide if a generalized Paley graph is suitable. Then
we used the tools referred to above and we presented conditions for two-
dimensional affine groups to be synchronizing.

iii
Resumo
Apropriedadedesincronizac¸˜aosurgiunocontextodateoriadeauto´matos
e semigrupos de transformac¸˜ao. Arnold and Steinberg definiram os grupos de
ˇ
permutac¸˜ao sincronizantes com o objectivo de estudar a Conjectura de Cerny´
por outra perspectiva. Nesta tese estudamos a propriedade de sincronizac¸˜ao
em grupos afins de pequenas dimenso˜es. Dado que J.E. Pin provou que os
gruposafinsdedimensa˜o1sa˜osempresincronizantes,osresultadosprincipais
desta tese aplicam-se a grupos afins bidimensionais.
Peter Neumann estudou os grupos sincronizantes usando teoria de grafos.
Esta caracteriza¸ca˜o consiste no estudo de grafos invariantes sobre a ac¸ca˜o
de um grupo e em determinar se os seus nu´meros croma´tico e de clique
coincidem. Descobrimosquealgunsgrafosinvariantessobreaac¸ca˜odegrupos
afins bidimensionais tinham uma geometria sim´etrica e que faziam parte de
uma classe amplamente estudada de grafos, conhecidos como grafos de Paley
generalizados.
Como ferramenta extra, estuda´mos a func¸˜ao-teta, um nu´mero invariante
num grafo, que foi definida por Lov´asz. Esta fun¸ca˜o possui uma caracteri-
za¸ca˜o que utiliza os valores pro´prios da matriz de adjacˆencia do grafo e que
nos permitiu concluir uma condi¸ca˜o suficiente para a igualdade no nu´mero
crom´atico e de clique em grafos cujo grupo de automorfismos´e transitivo nas
arestas e nos v´ertices.
Nestateseestabelecemosumcrit´erioparaaigualdadenonu´merocrom´atico
e de clique nos grafos de Paley generalizados e usa´mos esse resultado, bem
como as ferramentas anteriormente referidas, para obter uma caracteriza¸ca˜o
dos grupos afins bidimensionais sincronizantes.

v
Acknowledgment
I would like to express my gratitude to everyone that in one way or the
other was important for the elaboration of this dissertation. I apologize in
advance for the names that I might have forgotten to mention.
First of all, I want to present my deepest thanks to my supervisor Doutor
Csaba Schneider for all guidance and support during my Master’s degree.
Citing his Master’s thesis,
“( ...) the personality of the supervisor is of fundamental importance.”
I could not agree more. His enthusiasm with the research work has been fun-
damental for my motivation during this year. His new ideas and suggestions
for approaches to the problems increased my desire to further my knowledge
of mathematics. I am also thankful for his realistic perspective of the work
since he pushed me when it was necessary and he read carefully the awful
preliminary versions of the dissertation. His guidance in the research, during
courses and seminars and even his advices about giving classes, made me a
much better mathematician. Thank you very much.
I express my sincere gratitude to Joa˜o Arau´jo and Benjamin Steinberg for
theirhelponthehistoryofthesynchronizationproperty. Iwasverysurprised
by their willingness to answer my emails and explain to me the origin and
motivation for the study of the synchronization property on group theory.
Their contribution was fundamental to position synchronization across this
area of Algebra.
I would like to thank my professors from the mathematics department of
Faculdade de Ciˆencias, in particular, professor Owen Brisson and professor
Carlos Andr´e, for their availability for answering my doubts and questions
on group and number theory. Some of our discussions were the key for some
proofs in this dissertation. I also thank the support of the Center of Algebra
of the University of Lisbon, which allowed me to attend to several seminars
and workshops.
I also want to sincerely thank professors Gracinda Gomes and Maria
Jo˜ao Gouveia. However, it is not just enough for me to thank them for being
my teachers, since I am grateful to them for reasons beyond their academic
position. They are the ones who probably listened to most of my existential
problemswhilepursuingmyMaster’sandwho(alwaysfullofpatience)wisely
advised me. Their support and understanding were very important for me.
vi
I would also like to show my gratitude to all the people that, in spite of
their little understanding of the work of my dissertation, were an essential
support for me, in particular for my mental and psychological health. First,
I want to sincerely thank my mother and my grandparents for tolerating
my ill-mannered behavior and lack of patience in the low days and for their
infinite kindness and love.
I am happy to thank all my friends who have constantly been around,
caring about my work, celebrating my small victories and trying to cheer me
up when I was stuck with some problem. Thanks to Filipe, Marta, Raquel,
Mariana and Joana, the cute group, as they call. Thanks to Catarina and
Mafalda that, besides the jokes and parties for making me happy, taught me
that at some point we have to face the issues we are dealing with. Thanks
to Jeronimo for the interest in my work and also for encouraging me to have
higher prospects for the future, regardless of my fears. Thanks to Joana,
Marta, Catia, Beatriz and Tatah, the fancy girl’s group, for all the sharing
which was an important part of my emotional growth.
Finally, and trying to avoid a silly sentence, I deeply thank Ricardo for
being present all the time.