Combinatorial Group and Semigroup Theory

Code:

M502

Acronym:

M502

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	5	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

Getting acquaintance with the recent trends of combinatorial and geometric (semi)group theory.

Learning outcomes and competences

Capacity to solve exercises of advanced nature and ability to engage on research work in the field

Working method

Presencial

Program

1. Words and free monoids.

Words are the primordial object in both combinatorial group theory and combinatorial semigroup theory. We introduce their basic combinatorial properties and explore free monoids.

2. Languages and automata.

Languages are sets of words and they can be used to encode virtually everything, playing therefore a major role in theoretical computer science. Rational languages and finite automata are central.

3. Free groups.

Free groups play a major role in this course. We explore several important themes: Stallings' automata for finitely generated subgroups, the automorphism group, rational subsets, the boundary of the completion for the prefix metric.

4. Presentations.

This is the moment when we leave the safe coast of free objects for the open sea of (finite) presentations, where we subject free objects to a few defining relations and obtain therefore various types of quotients.

5. Van Kampen diagrams and pictures.

Van Kampen diagrams and pictures constitute the classical geometric tools to study group presentations, and have been successfully applied to study many types of presentations, such as in the case of small cancellation.

6. Free products and graph products.

We study free products and their generalization to the partial commutativity case: graph products. We start by the basic models of trace monoids and graph groups.

7. Hyperbolic groups.

Gromov introduces hyperbolic groups in 1987 with a revolutionary idea: to use the geometry of the Cayley graph to solve algorithmic problems. We study the properties and applications of the theory, including the central concept of quasi-isometry.

8. Automatic groups.

This concept can be defined through purely automata-theoretic features or blending it in alternative with a geometric property of the Cayley graph. We present some of its successes.

9. Self-similar groups.

Grigorchuk boosted in the last decade the study of self-similar groups, also known as automata groups. These are groups of sequential permutations built from a certain type of finite automata, and are deeply related to wreath products and actions in uniform trees of finite degree.

10. Free inverse monoids.

Inverse monoids correspond to partial injective transformations the way groups correspond to permutations. Free inverse monoids, in the beautiful Munn construction, are built from finite trees (finite automata, actually), and constitute undoubtedly one of the most aesthetic algebraic objects in Mathematics.

11. Inverse monoid presentations and Stephen's sequence.

Stephen's sequence emerges in the late eighties as a powerful tool to study various finite inverse monoid presentations from a geometric viewpoint. We introduce the construction and explore some of its nicest applications.

Mandatory literature

Pedro V. Silva; Teoria Geométrica de Grupos, 2014

Comments from the literature

Detailed bibliography can be provided upon request to the email pvsilva@fc.up.pt.

Teaching methods and learning activities

Exposition on blackboard. Discussion of exercises.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Trabalho escrito	50,00
Total:	100,00

Calculation formula of final grade

50% of the classification is obtained by the written resolution of 14 exercises. The other 50% are determined by the final exam, where the students are free to use notes to help them. A minimum marking of 8.5 is required in the final exam.