“You must attend the AMSI Summer School since it is highly
worth it. Not only you will enhance your knowledge and skills
in the area of mathematical sciences, but also you can broaden
your networking.”

Abdul Hadi Asfarangga, The University of Adelaide

Geometric Group Theory


Professor Murray Elder, University of Technology Sydney
Dr Adam Piggott, Australian National University
Dr Lawrence Reeves, The University of Melbourne
Associate Professor Anne Thomas, University of Sydney


Groups and geometry are ubiquitous in mathematics. This course will introduce students to the study of infinite groups from the geometrical viewpoint and will draw on ideas from low dimensional topology, hyperbolic geometry, and notions of self-similarity (fractal geometry). The principal focus is the interaction of geometry/topology and group theory: through group actions and suitable translations of geometric concepts into a group theoretic setting.

Course Overview

Topics covered in this course include:

  • Cayley graphs
  • Free groups and presentations, ping-pong lemma
  • Quasi-isometries, Milnor-Švarc lemma
  • Embedding theorems, Decision problems
  • Growth of groups
  • Dehn functions
  • Hyperbolic groups


Undergraduate abstract algebra (first groups & rings course at second/third year level). Desirable but not essential: metric spaces/introductory topology; discrete mathematics


  • Three written assignments 30% (one each weekend)
  • Special assignment 10%
  • Tutorial active contribution 5%
  • Final exam 55%

(Final assessment details to be confirmed)

Attendance requirements

  • For those completing the subject for their own knowledge/interest, final attendance requirements to be confirmed.

Resources/pre-reading (if available)


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Professor Murray Elder, University Of Technology Sydney

I am a pure mathematician working at the University of Technology Sydney. My research lies at the intersection of pure mathematics (algebra, geometric group theory, enumerative combinatorics) and theoretical computer science (computational complexity, algorithms, formal language theory). I am managing editor for the open access journal of Groups, Complexity, Cryptology.

Associate Professor Anne Thomas, University Of Sydney

Anne Thomas has a BA/BSc in History and Pure Mathematics from UNSW, and did her postgraduate work in Mathematics at the University of Chicago.  Since then, she has worked at Cornell, Oxford, Glasgow and Sydney.  Her research background is in geometric group theory and she frequently collaborates with mathematicians from other areas to study Coxeter groups and buildings from a geometric point of view.

Dr Adam Piggott, Australian National University

I am a pure mathematician at ANU. I think about combinatorial and geometric group theory, with applications to theoretical computer science.

Dr Lawrence Reeves, The University Of Melbourne

My research concentrates on geometric group theory. In particular, I am interested in automatic groups, hyperbolic groups, Coxeter groups CAT(0) groups and the relationships between these various classes.