96742 - Geometric Theory of Groups

Learning outcomes

At the end of the course, the student will have acquired the fundamentals of the theory of geometric group theory, and will be able to link algebraic properties of finitely generated groups and metric properties of spaces on which they act. They will be able to apply the acquired concepts to solve problems and construct proofs.

Course contents

Geometric Group Theory explores the deep connections between the geometric and algebraic properties of groups.

Central to this field are questions such as: Can groups be viewed as geometric objects, and how are their algebraic properties encoded in their geometry? More broadly, on which geometric spaces can specific groups act in a meaningful way, and how do the properties of these actions reflect the algebraic structure of the groups themselves?

For instance, the freeness of groups can be characterized via their actions on trees, providing a elegant proof that subgroups of free groups are themselves free. In this course, we will translate geometric concepts—such as geodesics, curvature, and volume—into the language of group theory.

The course is structured in two parts. The first part introduces free groups and the construction of Cayley graphs, providing a geometric framework for group representation. We will then examine the notion of quasi-isometry, which allows us to formalize the study of "large-scale geometry"—where two spaces are considered equivalent when viewed from a great distance (for example, a square grid and the Euclidean plane). This section culminates in the Švarc–Milnor Lemma, which establishes that a group acting "nicely" on a metric space is quasi-isometric to that space, thus sharing its large-scale geometric properties. We will conclude this part by investigating hyperbolic groups, which behave like hyperbolic space when observed from afar.

The second part of the course focuses on the growth of groups and its relationship with amenability. Growth is a quasi-isometric invariant; we will measure the volume of balls in a finitely generated group and analyze their asymptotic behavior as the radius tends to infinity. Amenable groups are characterized by the existence of a "mean"—a definition that, while analytic in nature, captures profound algebraic and geometric insights. We will prove that finitely generated groups with subexponential growth are necessarily amenable. The course concludes with an introduction to cohomological techniques, specifically providing a cohomological characterization of amenable groups through the vanishing of bounded cohomology and discussing the theory of quasimorphisms.

Ecco la traduzione in inglese, formulata con il tono appropriato per una sezione di un programma universitario:

Prerequisites: To fully appreciate the course content, it is advisable for students to have a solid background in algebra, geometry, and analysis, as typically covered during the first two years of a Bachelor's degree. In particular, familiarity with metric spaces, group theory, linear algebra, topology, and the fundamental group is recommended.

More advanced knowledge in geometry, topology, and analysis (such as algebraic topology, differential or Riemannian geometry, and functional analysis) is useful but not strictly required.

Readings/Bibliography

The course is based on the following books:

• Clara Löh - "Geometric group theory"
• Roberto Frigerio - "Bounded cohomology of discrete groups"

Other useful references for Geometric Group Theory are the following:

• Druţu, Kapovich - "Geometric group theory" (disponibile online [https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf] )
• Clay, Margalit - "Office Hours with a Geometric Group Theorist"
• Bridson, Haefliger - "Metric Spaces of Non-Positive Curvature"

Teaching methods

The course is organized in 48 hours of in-person teaching. Each lecture will contain new aspects of the theory as well as many examples/exercises. This will help the students to become more familiar with the new definitions.

Assessment methods

Every 4 lectures, an assignment sheet containing 2 exercises will be uploaded to the 'Virtuale' platform. A total of 6 sheets (12 exercises) will be provided. In preparation for the exam, students are required to prepare solutions for 4 exercises selected from 4 different sheets.

The exam consists of two parts. The first part (the theoretical exam) follows one of the two options below (with a maximum score of 22 points):

• Standard oral exam: Covers the entire course curriculum.

• Seminar-based oral exam: The seminar, lasting approximately 30 minutes (60 minutes for pairs, 90 minutes for groups of three), will focus on supplementary and in-depth topics related to the course material. A list of possible seminar topics will be uploaded to the 'Virtuale' platform at the beginning of the course. The seminar will be evaluated based on the following criteria: comprehension (5 points), exposition (5 points), organization of the seminar (5 points), use of appropriate technical terminology acquired during the course (5 points), and depth of analysis (2 points).

Following this, students will be asked to solve one of the 4 prepared exercises on the blackboard. Students will have 5 minutes to review their notes before presenting the solution. This part of the exam is worth a maximum of 10 points.

If the final total score is 31 or 32, the grade will be recorded as '30 e Lode'.

Teaching tools

Students can arrange meetings with the lecturers (scheduled in advance via email) in order to ask questions about the theory as well as about the topics for the seminar and the exercises.

Students with learning disabilities (LD) or temporary or permanent disabilities are advised to promptly contact the relevant University office at https://site.unibo.it/studenti-con-disabilita-e-dsa/en . The office will recommend some adjustments, which must be submitted to the lecturer for approval 15 days in advance. The lecturer will evaluate their compatibility, taking into account the course's learning goals.

Office hours

See the website of Marco Moraschini

See the website of Maria Beatrice Pozzetti