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Young Geometric Group Theory
Provider: Faculty of Science

Activity no.: 7013-25-07-31
Enrollment deadline: 07/04/2025
PlaceDepartment of Geoscience and Natural Resource Management
Date and time07.04.2025, at: 09:00 - 11.04.2025, at: 16:00
Regular seats200
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserDamian Longin Osajda    E-mail address: do@math.ku.dk
Teaching languageEnglish
Exam formContinuous assessment
Course workload
Course workload categoryHours
Preparation45.00
Lectures12.00
Theory exercises12.00
Sum69.00
Aim and content
The Specialised PhD Course consists of three lecture series accompanied by exercises, the topics of which are the following:

“Free-by-cyclic groups” by Radhika Gupta: This course will be devoted to the study of free-by-cyclic groups. Free-by-cyclic groups, which are a graphical analogue of fundamental groups of three-manifolds fibering over the circle, are a rich class of groups with interesting and varied properties. The goal of this workshop is to spark interest and build expertise in problems related to free-by-cyclic groups that are motivated by results in the study of fibered three-manifolds.

“Helly graphs and coarse nonpositive curvature” by Thomas Hattel: The course will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. It will focus on nonpositive features, such as local properties, bicombings and classification of isometries. It will present various rich constructions of such spaces, quite often associated with groups of geometric nature, such as hyperbolic groups, braid groups, mapping class groups and higher rank lattices.

“Finiteness properties of groups” by Brita Nucinkis: Finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. They are algebraic and geometric notions that generalise the properties of being finitely generated and finitely presented. This course will introduce several of these properties and provide examples of how they appear and are used in geometric group theory.

Formal requirements
Basic knowledge of group theory, topology and metric spaces.

Learning outcome
Knowledge:
* Modern notions of coarse and combinatorial non-positive curvature
* Theory of free-by-cyclic groups
* Finiteness properties of groups and their applications
* Understanding of today’s mainstream topics in Geometric Group Theory (GGT) and related fields
* Knowledge about the international community of GGT researchers, including present and future important centers and programs (semesters etc)

Skills:
* Problem solving and recent proof techniques
* Communication skills (participating in discussions and disseminating results)
* Networking skills inside the Geometric Group Theory (GGT) community and beyond (including participation in related programs and projects, eg grants)

Competences:
* Identifying and valuing research tasks within today’s Geometric Group Theory (GGT)
* Initiating and performing collaboration on projects with researchers within GGT and beyond
* Planning and carrying out research and development tasks in various contexts

Literature
References for “Free-by-cyclic groups”:
• S. Dowdall, R. Gupta, S. J. Taylor, Orientable maps and polynomial invariants of free-by-cyclic groups, Advances in Mathematics, 415, 15 February 2023
• O. Bogopolski, A. Martino, and E. Ventura. The automorphism group of a free-by-cyclic group in rank 2. Comm. Algebra, 35(5):1675–1690, 2007

References for “Helly graphs and coarse nonpositive curvature”:
• T. Haettel, Group actions on injective spaces and Helly graphs, https://arxiv.org/abs/2307.00414, 2023.
• U. Lang, Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal., 5(3):297–331, 2013.

References for “Finiteness properties of groups”:
• M. Bestvina and N. Brady, Morse theory and finiteness properties of groups. Inventiones Mathematicae, 129 (3): 445–470, 1997.
• I.J. Leary, B. Nucinkis, Some groups of type VF, Inventiones Math. 151 (1) (2003), 135-165.
• K.S. Brown, Cohomology of groups. Springer-Verlag, 1982.

Target group
MSc and PhD students and postdocs working on Geometric Group Theory.

Teaching and learning methods
Each course will consist of four one hour lectures and discussion sessions where the participants will be able to solve problems and interact directly with leading mathematicians. Written material will be provided for each course, so that the participants can prepare and deepen their knowledge on the topics.

Lecturers
Guest lecturers:
Radhika Gupta, Tata Institute of Fundamental Research, India
Thomas Haettel, University of Montpellier, France
Brita Nucinkis, University of London, UK

Remarks
This specialized PhD course will be part of the larger conference/school Young Geometric Group Theory XIII and its sign up requirements are the same.
Registration will open i the fall 2024.
The deadline for registration will be in February 2025.

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