An Introduction to Hyperbolic Geometry

By Prof. Abhijit Pal   |   IIT Kanpur
Learners enrolled: 102
Hyperbolic geometry is a non-Euclidean Geometry which has wide applications in mathematics and is a central object to study group theory from a geometrical view point. Many surfaces and three manifolds exhibit hyperbolic geometry. In this course, we will learn basic hyperbolic geometry and then move on to its applications in geometric group theory.

INTENDED AUDIENCE: PG & PhD Students but an UG student with background in topology, algebra and complex analysis can take this course.

PREREQUISITES: Topology, Algebra (Group Theory), Complex Analysis
Course Status : Upcoming
Course Type : Elective
Duration : 12 weeks
Start Date : 23 Jan 2023
End Date : 14 Apr 2023
Exam Date : 30 Apr 2023 IST
Enrollment Ends : 30 Jan 2023
Category :
  • Mathematics
Credit Points : 3
Level : Postgraduate

Page Visits

Course layout

Week 1-5 :  Hyperbolic Geometry : Models of Hyperbolic space: Upper half plane and Unit Disc with Poincaré metric, Hyperbolic Inner Product, Geodesics, Isometry Groups, Classification of Isometries, Area of Triangles, Trignometric Identities. 

Week 6-8 : Properly discontinous action, Cocompact action, Fuchsian Group, Covering Space, Fundamental Region, Tesellation, Algebraic properties of Fuchsian Group, Hyperbolic surfaces, Poincaré’s Theorem (Statement Only) Closed curves and geodesics on Hyperbolic Surfaces .

Week 9-12 : Slim and Thin triangles, Hyperbolic metric space, Exponential Divergence, Stability of Quasi-geodesics. Hyperbolic groups and its properties. 

Books and references

Svetlana Katok: Fuchsian Groups, The University of Chicago Press.
M. R. Bridson and A. Haefliger: Metric Spaces of Non-Positive Curvature, Springer.
B. H. Bowditch: A course on geometric group theory,

Instructor bio

Prof. Abhijit Pal

IIT Kanpur
Prof. Abhijit Pal is a faculty member in the Department of Mathematics & Statistics of IIT-Kanpur. He has done PhD from ISI-Kolkata in 2011 under the supervision of Prof. Mahan Mj. His research interest lies in Hyperbolic Geometry, Geometric Group Theory, specially in hyperbolic groups and relatively hyperbolic groups.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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