Summer Graduate School

Rips's δ-thin triangle condition for Gromov hyperbolicity of metric spaces (Stomatapoll)

Geometric group theory studies discrete groups by understanding the connections between algebraic properties of these groups and topological and geometric properties of the spaces on which they act. The aim of this summer school is to  introduce graduate students to specific central topics and recent developments in geometric group theory. The school will also include students presentations to give the participants an opportunity to practice their speaking skills in mathematics.  Finally, we hope that this meeting will help connect Latin American students with their American and Canadian counterparts in an environment that encourages discussion and collaboration. 

The first week of the school will be devoted to group actions on Hilbert spaces, orderable groups and dynamics in dimension 1, with mini-courses by T. Fernós and A. Navas. The second week of the school will be dedicated to two important sources of examples in geometric group theory: mapping class groups and groups of outer automorphisms of free groups. Connections with low dimensional topology and hyperbolic groups will be explored. This second week will feature mini-courses by J. Tao and S. Dowdall. 

For eligibility and how to apply, see the Summer Graduate Schools homepage

Suggested Prerequisites:

Students should have basic knowledge of calculus, mathematical analysis and measure theory (knowing Hilbert space theory and lp spaces will be enough). They are also expected to know the standard topics in abstract algebra (linear algebra, groups, rings, fields) with an emphasis on group theory. Other relevant topics include differential geometry and topology and algebraic topology (one course on differential geometry and abstract manifolds and knowing the content of chapter 1 of Hatcher's book "Algebraic Topology" is enough). Furthermore, having taken a basic course on dynamical systems could be useful (yet this is not mandatory).

Finally, as a general reference on the topics of the school that the students could browse, we mention Drutu and Kapovich's book Geometric Group Theory, specifically chapters 4, 5, 7, 8 and 11 (this book was published by the AMS in 2018, it is available in preprint form from https://www.math.ucdavis.edu/%7Ekapovich/papers.html). Further references are Farb and Margalit's book A primer on Mapping Class Group (for instance chapters 1, 2 and 3), the lectures notes Notes on word hyperbolic groups by Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro and Short (in Group theory from a geometrical viewpoint (Trieste, 1990)), the book by Deroin, Navas and Rivas, Groups, Orders and Dynamics (to be published by the AMS, available in preprint form at https://arxiv.org/abs/1408.5805), specifically its first chapter and finally the two standard references on group actions on Hilbert spaces (Kazhdan's property (T), by Bekka, de La Harpe and Valette and Groups with the Haagerup property: Gromov's a-T-menability, by Cherix, Cowling, Jolissaint, Julg and Valette). These references would hopefully give a preview of some of the material that will be covered and will help the students to take more advantage of the school.

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.