Geometric Group Theory

Lecturers

Dr Alejandra Garrido Angulo, The University of Newcastle
Associate Professor Murray Elder, University of Technology, Sydney
Dr Lawrence Reeves, The University of Melbourne

Synopsis

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

  • Free groups and presentations, ping-pong lemma
  • Cayley graphs, quasi-isometries, Milnor-Švarc lemma
  • Hyperbolic groups
  • Embedding theorems, Decision problems
  • Nilpotent groups, soluble groups
  • Growth of groups
  • Self-similar groups

Prerequisites

  • Undergraduate abstract algebra (first groups & rings course)
  • Discrete maths/combinatorics/proofs course
  • Desirable: metric spaces/topology course

Assessment

  • Three written assignments 30%
  • Present problem in small groups 5%
  • Special assignment 10%
  • Final exam 55%

Resources/pre-reading (if available)

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Dr Alejandra Garrido Angulo,
The University of Newcastle

I am interested in infinite groups, and study them both algebraically and by viewing them as symmetries of geometric or topological objects. My research is in groups of (rooted) tree automorphisms and their relatives of dynamical origin, profinite groups, and totally disconnected locally compact groups.

I am currently a postdoctoral research associate in the Zero-Dimensional Symmetry group led by George Willis at the University of Newcastle, since Oct 2018. Before that, I had postdocs at Düsseldorf and Geneva. I defended my DPhil in 2015 under the supervision of John Wilson.

Dr Lawrence Reeves,
The University of Melbourne

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

Associate Professor Murray Elder,
University of Technology, Sydney

I am a pure mathematician working at the University of Technology Sydney. My research interests include geometric group theory, complexity theory, automata and formal language theory, enumerative combinatorics and pattern-avoiding permutations. I am an editor for Groups, Complexity, Cryptology, and a current member of the Australian Research Council College of Experts.