word problem (for groups)
Suppose someone hands you a group by a presentation and writes down a string of generators and inverses — a word. You want to know one apparently simple thing: does this word, multiplied out, equal the identity? Equivalently, does a given product of symbols collapse to nothing once you apply the group's relations? That yes-or-no question, asked uniformly for all words in a fixed presentation, is the word problem.
Made precise, the word problem for a finitely presented group G asks for an algorithm that, given any word w in the generators, decides whether w = 1 in G. It is one of three classical decision problems Max Dehn isolated in 1911, alongside the conjugacy problem and the isomorphism problem. A group is said to have solvable word problem when such an algorithm exists; this is a property of the group, not of any one presentation, since changing presentations is itself algorithmic.
The startling fact, proved independently by Novikov (1955) and Boone (1958), is that there exist finitely presented groups with unsolvable word problem: no algorithm can decide identity for them. So the word problem is undecidable in general. Yet enormous classes solve it positively — finite groups, free groups, abelian and nilpotent groups, hyperbolic groups, automatic groups, residually finite finitely presented groups — and geometric group theory largely measures how hard the problem is via the Dehn function.
In the free group F(a, b) the word abab^(-1)a^(-1)b^(-1) is not the identity (it is already reduced and nonempty), while abb^(-1)a^(-1) reduces to the empty word and so equals 1. Free reduction solves the word problem here trivially.
In a free group, just freely reduce; in general no such easy test need exist.
Solvability of the word problem is invariant under quasi-isometry among finitely presented groups, one reason geometric methods bear on this purely algebraic question.