The Andrews Curtis Conjecture Explained: Free Groups, AC Moves & the AC Challenge

Ever wondered why the trivial group, a group with just one element, could hide an open problem that has resisted mathematicians since 1965? In this session we break down the Andrews–Curtis conjecture from scratch and look at how researchers are attacking it with search algorithms and reinforcement learning. We start with the basics: free groups, generators and relators, and balanced presentations of the trivial group. Then we define the three Andrews–Curtis (AC) moves (inversion, multiplication and conjugation) and work through a training example by hand. After that we cover: The AC Challenge: 10,000+ two-generator problems and how to read the competition’s JSON format Famous potential counterexamples: the Akbulut–Kirby family (including AK(3)) and the Miller–Schupp family Breadth-first search vs greedy search, and why the search space explodes exponentially Reinforcement learning (PPO) results and the “two-hump” difficulty problem Supermoves and substitution, and how they cut a million-node search down to a few hundred Open directions where you could contribute, and papers to read next I also share my own results from the challenge: greedy search solved the most problems for me, with breadth-first search close behind. 📄 Papers mentioned: Andrews & Curtis (1965), Free Groups and Handlebodies The Caltech “two-hump” paper on AC presentations and reinforcement learning The 2025 paper on what makes math problems hard for reinforcement learning (BFS, greedy search, PPO, Miller–Schupp subfamilies) The enumeration of balanced presentations of the trivial group up to total length 13 If you’re into group theory, combinatorial topology, or AI for math, this is a great problem to get your hands dirty with. Questions are welcome in the comments! Chapters 00:00 Introduction • What the Andrews–Curtis conjecture is about • Why AI and reinforcement learning researchers care 00:54 Group Theory Basics • Many ways to write the same group • Free groups • Generators and relators • Presentations of the trivial group 10:25 The Andrews–Curtis Conjecture • Statement of the conjecture • The three AC moves • Worked example: solving a training problem by hand 19:53 The AC Challenge • How the competition works • Reading the problem files • The 14 moves used for verification • Recap: balanced presentations and Nielsen transformations 24:48 Potential Counterexamples • The Akbulut–Kirby family • The Miller–Schupp family • AK(3): a possible shortest counterexample [timestamp] Search and AI Approaches • Breadth-first and greedy search • Reinforcement learning and the two-hump problem • Supermoves and substitution [timestamp] Why It Matters and Where to Start • Why RL researchers find this problem interesting • Papers to read • Open research directions Tags Andrews-Curtis conjecture, AC conjecture, group theory, free groups, combinatorial group theory, trivial group, group presentations, AC moves, Nielsen transformations, Akbulut-Kirby, Miller-Schupp, reinforcement learning, AI for math, PPO, greedy search, breadth first search, open problems in mathematics, algebraic topology, math competition, AC challenge Hashtags #GroupTheory #AndrewsCurtis #ReinforcementLearning #AIforMath #Mathematics