Andrews–Curtis conjecture

Think of letters as reversible instructions: a means do an action, and $a^{-1}$ means undo it. A relation declares that a particular string does nothing. For example, the two rules $a=1$ and $ab=1$ force both a and b to do nothing; 1 denotes doing nothing.

Some much longer rule lists also collapse every instruction to doing nothing. When there are equally many letters and relations, can a fixed set of rule changes always expose this simplicity? Andrews–Curtis asks this for every such description of the trivial group. The changes may reverse a relation, join two relations, or surround a relation by an instruction and its undo, with ordinary adjacent cancellations. No extra letters may be introduced.[1]

A finite list of reversible instructions may have rules that make every possible string equivalent to doing nothing. Can that complicated description always be rewritten into one rule saying that each instruction does nothing, using only the Andrews–Curtis moves? The question assumes equal numbers of instructions and rules and allows no temporary extra instruction.

Write a generator a for an instruction, $a^{-1}$ for its undo, and 1 for doing nothing. A relation is a word declared equal to 1. The presentation $\langle a,b\mid a,ab\rangle$ is trivial: a is already 1, so ab equal to 1 also forces b equal to 1. Its legal relation rewrites are $$(a,ab)\longrightarrow(a^{-1},ab)\longrightarrow(a^{-1},a^{-1}ab)\longrightarrow(a^{-1},b)\longrightarrow(a,b).$$ The steps are inversion, multiplication by another relation on the left, adjacent inverse cancellation, and inversion.

In general, a balanced presentation has n generators and n relations. The allowed moves invert a relation, multiply it on either side by another relation, or conjugate it: surround it by a generator and its inverse, as in $r\mapsto ara^{-1}$. Inserting or removing adjacent inverse pairs is also allowed. The target is the list of the original generators themselves. Stabilization would add a new generator together with the rule that it is 1; that belongs to a different version of the question.[1]

Selected classes and individual examples can be simplified. The permitted moves have not been shown to work for every such presentation.[1]

The June 2026 preprint proves the ordinary conjecture for thickenable presentations. Stable versions allow adding generator–relation pairs and have a different move set.[1]

A symbolic rule list can also describe a space: letters give loops and relations attach disks that fill specified loops. Whether loops can shrink is then encoded by the group. The question connects simplifying equations with simplifying these descriptions of spaces, where an apparently tangled construction can have very simple underlying behavior.[1]

A group presentation has a concrete topological realization: start with a point, attach a loop for every generator, and attach a disk along the loop path spelled out by every relation. The resulting space has the presented group as its fundamental group, which records loops up to continuous shrinking and deformation.

The conjecture studies how this topological simplicity is reflected in restricted rewrites of the disk instructions. The June preprint uses presentations whose associated space embeds in a three-dimensional manifold, making this connection part of an actual proof for a special class. The broader claim does not itself supply an algorithm to decide whether an arbitrary presentation is trivial.[1]